Electronic Journal of Probability Articles (Project Euclid)
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Lévy Classes and Self-Normalization
http://projecteuclid.org/euclid.ejp/1453756464
<strong>Davar Khoshnevisan</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 1, 18 pp..</p><p><strong>Abstract:</strong><br/>
We prove a Chung's law of the iterated logarithm for recurrent linear Markov processes. In order to attain this level of generality, our normalization is random. In particular, when the Markov process in question is a diffusion, we obtain the integral test corresponding to a law of the iterated logarithm due to Knight.
</p>projecteuclid.org/euclid.ejp/1453756464_20160125161428Mon, 25 Jan 2016 16:14 ESTRapid social connectivityhttps://projecteuclid.org/euclid.ejp/1554775411<strong>Itai Benjamini</strong>, <strong>Jonathan Hermon</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 33 pp..</p><p><strong>Abstract:</strong><br/>
Given a graph $G=(V,E)$, consider Poisson($|V|$) walkers performing independent lazy simple random walks on $G$ simultaneously, where the initial position of each walker is chosen independently with probability proportional to the degrees. When two walkers visit the same vertex at the same time they are declared to be acquainted. The social connectivity time $\mathrm{SC} (G)$ is defined as the first time in which there is a path of acquaintances between every pair of walkers. It is shown that when the average degree of $G$ is $d$, with high probability \[ c\log |V| \le \mathrm{SC} (G) \le C d^{1+5 \cdot 1_{G \text{ is not regular} } } \log ^3 |V|. \] When $G$ is regular the lower bound is improved to $\mathrm{SC} (G) \ge \log |V| -6 \log \log |V| $, with high probability. We determine $\mathrm{SC} (G)$ up to a constant factor in the cases that $G$ is an expander and when it is the $n$-cycle.
</p>projecteuclid.org/euclid.ejp/1554775411_20190704220503Thu, 04 Jul 2019 22:05 EDTContinuous-state branching processes with competition: duality and reflection at infinityhttps://projecteuclid.org/euclid.ejp/1554775412<strong>Clément Foucart</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 38 pp..</p><p><strong>Abstract:</strong><br/>
The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty $ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty $ is inaccessible, it is always an entrance boundary. In the case where $\infty $ is accessible, explosion can occur either by a single jump to $\infty $ (the process at $z$ jumps to $\infty $ at rate $\lambda z$ for some $\lambda >0$) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when $\infty $ is accessible and $0\leq \frac{2\lambda } {c}<1$, the extended process is reflected at $\infty $. In the case $\frac{2\lambda } {c}\geq 1$, $\infty $ is an exit of the extended process. When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at $\infty $ gets extinct almost surely. Moreover absorption at $0$ is almost sure if and only if Grey’s condition is satisfied. When the branching mechanism is the Laplace exponent of a subordinator, necessary and sufficient conditions are given for a stationary distribution to exist. The Laplace transform of the latter is provided. The study is based on classical time-change arguments and on a new duality method relating logistic CSBPs with certain generalized Feller diffusions.
</p>projecteuclid.org/euclid.ejp/1554775412_20190704220503Thu, 04 Jul 2019 22:05 EDTDistances between zeroes and critical points for random polynomials with i.i.d. zeroeshttps://projecteuclid.org/euclid.ejp/1554775413<strong>Zakhar Kabluchko</strong>, <strong>Hauke Seidel</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 25 pp..</p><p><strong>Abstract:</strong><br/>
Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi _0,\xi _1,\ldots ,\xi _n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to \infty $, with high probability there is a critical point of $Q_n$ which is very close to $\xi _0$. We localize the position of this critical point by proving that the difference between $\xi _0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\xi _0))$ and variance of order $\log n \cdot n^{-3}$. Here, $f(z)= \mathbb E [\frac 1 {z-\xi _k}]$ is the Cauchy–Stieltjes transform of the $\xi _k$’s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.
</p>projecteuclid.org/euclid.ejp/1554775413_20190704220503Thu, 04 Jul 2019 22:05 EDTWasserstein-2 bounds in normal approximation under local dependencehttps://projecteuclid.org/euclid.ejp/1554775414<strong>Xiao Fang</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 14 pp..</p><p><strong>Abstract:</strong><br/>
We obtain a general bound for the Wasserstein-2 distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. We apply the main result to obtain Wasserstein-2 bounds in normal approximation for sums of $m$-dependent random variables, U-statistics and subgraph counts in the Erdős-Rényi random graph. We state a conjecture on Wasserstein-$p$ bounds for any positive integer $p$ and provide supporting arguments for the conjecture.
</p>projecteuclid.org/euclid.ejp/1554775414_20190704220503Thu, 04 Jul 2019 22:05 EDTRescaled Whittaker driven stochastic differential equations converge to the additive stochastic heat equationhttps://projecteuclid.org/euclid.ejp/1554775415<strong>Yu-Ting Chen</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 33 pp..</p><p><strong>Abstract:</strong><br/>
We study some linear SDEs arising from the two-dimensional $q$-Whittaker driven particle system on the torus as $q\to 1$. The main result proves that the SDEs along certain characteristics converge to the additive stochastic heat equation. Extensions for the SDEs with generalized coefficients and in other spatial dimensions are also obtained. Our proof views the limiting process after recentering as a process of the convolution of a space-time white noise and the Fourier transform of the heat kernel. Accordingly we turn to similar space-time stochastic integrals defined by the SDEs, but now the convolution and the Fourier transform are broken. To obtain tightness of these induced integrals, we bound the oscillations of complex exponentials arising from divergence of the characteristics, with two methods of different nature.
</p>projecteuclid.org/euclid.ejp/1554775415_20190704220503Thu, 04 Jul 2019 22:05 EDTConfinement of Brownian polymers under geometric area tiltshttps://projecteuclid.org/euclid.ejp/1554775416<strong>Pietro Caputo</strong>, <strong>Dmitry Ioffe</strong>, <strong>Vitali Wachtel</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
We consider confinement properties of families of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. The model is introduced in order to mimic level lines of $2+1$ discrete Solid-On-Solid random interfaces above a hard wall.
</p>projecteuclid.org/euclid.ejp/1554775416_20190704220503Thu, 04 Jul 2019 22:05 EDTRandom walk in cooling random environment: ergodic limits and concentration inequalitieshttps://projecteuclid.org/euclid.ejp/1554775418<strong>Luca Avena</strong>, <strong>Yuki Chino</strong>, <strong>Conrado da Costa</strong>, <strong>Frank den Hollander</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 35 pp..</p><p><strong>Abstract:</strong><br/>
In previous work by Avena and den Hollander [3], a model of a random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a given sequence of times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime , a weak law of large numbers and certain fluctuation properties were derived under the annealed measure, in dimension one. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the random walk can be represented as a sum of independent variables, distributed as the increments of a random walk in a static random environment over diverging periods of time. Our proofs require suitable multi-layer decompositions of sums of random variables controlled by moment bounds and concentration estimates. Along the way we derive two results of independent interest, namely, concentration inequalities for the random walk in the static random environment and an ergodic theorem that deals with limits of sums of triangular arrays representing the structure of the cooling regime. We close by discussing our present understanding of homogenisation effects as a function of the cooling scheme, and by hinting at what can be done in higher dimensions. We argue that, while the cooling scheme does not affect the speed in the strong law of large numbers nor the rate function in the large deviation principle, it does affect the fluctuation properties.
</p>projecteuclid.org/euclid.ejp/1554775418_20190704220503Thu, 04 Jul 2019 22:05 EDTAnnealed scaling relations for Voronoi percolationhttps://projecteuclid.org/euclid.ejp/1554861841<strong>Hugo Vanneuville</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 71 pp..</p><p><strong>Abstract:</strong><br/>
We prove annealed scaling relations for planar Voronoi percolation. To our knowledge, this is the first result of this kind for a continuum percolation model. We are mostly inspired by the proof of scaling relations for Bernoulli percolation by Kesten [22]. Along the way, we show an annealed quasi-multiplicativity property by relying on the quenched box-crossing property proved by Ahlberg, Griffiths, Morris and Tassion [3]. Intermediate results also include the study of quenched and annealed notions of pivotal events and the extension of the quenched box-crossing property of [3] to the near-critical regime.
</p>projecteuclid.org/euclid.ejp/1554861841_20190704220503Thu, 04 Jul 2019 22:05 EDTFluctuation theory for Lévy processes with completely monotone jumpshttps://projecteuclid.org/euclid.ejp/1555034439<strong>Mateusz Kwaśnicki</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 40 pp..</p><p><strong>Abstract:</strong><br/>
We study the Wiener–Hopf factorization for Lévy processes $X_{t}$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener–Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_{t}$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_{t}$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi )$ of $X_{t}$, including a peculiar structure of the curve along which $f(\xi )$ takes real values.
</p>projecteuclid.org/euclid.ejp/1555034439_20190704220503Thu, 04 Jul 2019 22:05 EDTErgodicity of some classes of cellular automata subject to noisehttps://projecteuclid.org/euclid.ejp/1555034440<strong>Irène Marcovici</strong>, <strong>Mathieu Sablik</strong>, <strong>Siamak Taati</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 44 pp..</p><p><strong>Abstract:</strong><br/>
Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise.
We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.
</p>projecteuclid.org/euclid.ejp/1555034440_20190704220503Thu, 04 Jul 2019 22:05 EDTHarmonic functions on mated-CRT mapshttps://projecteuclid.org/euclid.ejp/1561082667<strong>Ewain Gwynne</strong>, <strong>Jason Miller</strong>, <strong>Scott Sheffield</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 55 pp..</p><p><strong>Abstract:</strong><br/>
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $\gamma $-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$ if we take the correlation to be $-\cos (\pi \gamma ^{2}/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties of random walk and discrete conformal embeddings for these maps.
For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps — including mated-CRT maps and the UIPT — the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_{n}(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor.
</p>projecteuclid.org/euclid.ejp/1561082667_20190704220503Thu, 04 Jul 2019 22:05 EDTThe non-linear sewing lemma I: weak formulationhttps://projecteuclid.org/euclid.ejp/1561082668<strong>Antoine Brault</strong>, <strong>Antoine Lejay</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 24 pp..</p><p><strong>Abstract:</strong><br/>
We introduce a new framework to deal with rough differential equations based on flows and their approximations. Our main result is to prove that measurable flows exist under weak conditions, even if solutions to the corresponding rough differential equations are not unique. We show that under additional conditions of the approximation, there exists a unique Lipschitz flow. Then, a perturbation formula is given. Finally, we link our approach to the additive, multiplicative sewing lemmas and the rough Euler scheme.
</p>projecteuclid.org/euclid.ejp/1561082668_20190704220503Thu, 04 Jul 2019 22:05 EDTRandom field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noiseshttps://projecteuclid.org/euclid.ejp/1561082669<strong>Robert C. Dalang</strong>, <strong>Thomas Humeau</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 28 pp..</p><p><strong>Abstract:</strong><br/>
We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise, with symmetric $\alpha $-stable Lévy white noise as an important special case. We identify conditions for existence of these two kinds of solutions, and, together with a new stochastic Fubini theorem, we provide conditions under which they are essentially equivalent. We apply these results to the linear stochastic heat, wave and Poisson equations driven by a symmetric $\alpha $-stable Lévy white noise.
</p>projecteuclid.org/euclid.ejp/1561082669_20190704220503Thu, 04 Jul 2019 22:05 EDTConvergence of the population dynamics algorithm in the Wasserstein metrichttps://projecteuclid.org/euclid.ejp/1561082670<strong>Mariana Olvera-Cravioto</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 27 pp..</p><p><strong>Abstract:</strong><br/>
We study the convergence of the population dynamics algorithm, which produces sample pools of random variables having a distribution that closely approximates that of the special endogenous solution to a variety of branching stochastic fixed-point equations, including the smoothing transform, the high-order Lindley equation, the discounted tree-sum and the free-entropy equation. Specifically, we show its convergence in the Wasserstein metric of order $p$ ($p \geq 1$) and prove the consistency of estimators based on the sample pool produced by the algorithm.
</p>projecteuclid.org/euclid.ejp/1561082670_20190704220503Thu, 04 Jul 2019 22:05 EDTMarkov chains with heavy-tailed increments and asymptotically zero drifthttps://projecteuclid.org/euclid.ejp/1561082671<strong>Nicholas Georgiou</strong>, <strong>Mikhail V. Menshikov</strong>, <strong>Dimitri Petritis</strong>, <strong>Andrew R. Wade</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 28 pp..</p><p><strong>Abstract:</strong><br/>
We study the recurrence/transience phase transition for Markov chains on ${\mathbb{R} }_{+}$, $\mathbb{R} $, and ${\mathbb{R} }^{2}$ whose increments have heavy tails with exponent in $(1,2)$ and asymptotically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On ${\mathbb{R} }_{+}$, for example, we show that if the tail of the positive increments is about $c y^{-\alpha }$ for an exponent $\alpha \in (1,2)$ and if the drift at $x$ is about $b x^{-\gamma }$, then the critical regime has $\gamma = \alpha -1$ and recurrence/transience is determined by the sign of $b + c\pi \operatorname{cosec} (\pi \alpha )$. On $\mathbb{R} $ we classify whether transience is directional or oscillatory, and extend an example of Rogozin & Foss to a class of transient martingales which oscillate between $\pm \infty $. In addition to our recurrence/transience results, we also give sharp results on the existence/non-existence of moments of passage times.
</p>projecteuclid.org/euclid.ejp/1561082671_20190704220503Thu, 04 Jul 2019 22:05 EDTNon local branching Brownian motions with annihilation and free boundary problemshttps://projecteuclid.org/euclid.ejp/1561082672<strong>Anna De Masi</strong>, <strong>Pablo A. Ferrari</strong>, <strong>Errico Presutti</strong>, <strong>Nahuel Soprano-Loto</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 30 pp..</p><p><strong>Abstract:</strong><br/>
We study a system of branching Brownian motions on $\mathbb{R} $ with annihilation: at each branching time a new particle is created and the leftmost one is deleted. The case of strictly local creations (the new particle is put exactly at the same position of the branching particle) was studied in [10]. In [11] instead the position $y$ of the new particle has a distribution $p(x,y)dy$, $x$ the position of the branching particle, however particles in between branching times do not move. In this paper we consider Brownian motions as in [10] and non local branching as in [11] and prove convergence in the continuum limit (when the number $N$ of particles diverges) to a limit density which satisfies a free boundary problem when this has classical solutions. We use in the convergence a stronger topology than in [10] and [11] and have explicit bounds on the rate of convergence.
</p>projecteuclid.org/euclid.ejp/1561082672_20190704220503Thu, 04 Jul 2019 22:05 EDTExistence of a phase transition of the interchange process on the Hamming graphhttps://projecteuclid.org/euclid.ejp/1561169148<strong>Piotr Miłoś</strong>, <strong>Batı Şengül</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
The interchange process on a finite graph is obtained by placing a particle on each vertex of the graph, then at rate $1$, selecting an edge uniformly at random and swapping the two particles at either end of this edge. In this paper we develop new techniques to show the existence of a phase transition of the interchange process on the $2$-dimensional Hamming graph. We show that in the subcritical phase, all of the cycles of the process have length $O(\log n)$, whereas in the supercritical phase a positive density of vertices lies in cycles of length at least $n^{2-\varepsilon }$ for any $\varepsilon >0$.
</p>projecteuclid.org/euclid.ejp/1561169148_20190704220503Thu, 04 Jul 2019 22:05 EDTSpatial moments for high-dimensional critical contact process, oriented percolation and lattice treeshttps://projecteuclid.org/euclid.ejp/1561169149<strong>Akira Sakai</strong>, <strong>Gordon Slade</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 18 pp..</p><p><strong>Abstract:</strong><br/>
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We use the lace expansion to prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.
</p>projecteuclid.org/euclid.ejp/1561169149_20190704220503Thu, 04 Jul 2019 22:05 EDTFirst passage time of the frog model has a sublinear variancehttps://projecteuclid.org/euclid.ejp/1562292237<strong>Van Hao Can</strong>, <strong>Shuta Nakajima</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 27 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we show that the first passage time in the frog model on $\mathbb{Z} ^{d}$ with $d\geq 2$ has a sublinear variance. This implies that the central limit theorem does not hold at least with the standard diffusive scaling. The proof is based on the method introduced in [4, 11] combined with a control of the maximal weight of paths in a locally dependent site-percolation. We also apply this method to get the linearity of the lengths of optimal paths.
</p>projecteuclid.org/euclid.ejp/1562292237_20190704220503Thu, 04 Jul 2019 22:05 EDTQuantitative CLTs for symmetric $U$-statistics using contractionshttps://projecteuclid.org/euclid.ejp/1549681361<strong>Christian Döbler</strong>, <strong>Giovanni Peccati</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 43 pp..</p><p><strong>Abstract:</strong><br/>
We consider sequences of symmetric $U$-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of contraction operators . Our results represent an explicit counterpart to analogous criteria that are available for sequences of random variables living on the Gaussian, Poisson or Rademacher chaoses, and are perfectly tailored for geometric applications. As a demonstration of this fact, we develop explicit bounds for subgraph counting in generalised random graphs on Euclidean spaces; special attention is devoted to the so-called ‘dense parameter regime’ for uniformly distributed points, for which we deduce CLTs that are new even in their qualitative statement, and that substantially extend classical findings by Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).
</p>projecteuclid.org/euclid.ejp/1549681361_20190716040113Tue, 16 Jul 2019 04:01 EDTBehavior of the empirical Wasserstein distance in ${\mathbb R}^d$ under moment conditionshttps://projecteuclid.org/euclid.ejp/1550113245<strong>Jérôme Dedecker</strong>, <strong>Florence Merlevède</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 32 pp..</p><p><strong>Abstract:</strong><br/>
We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order $p\in [1, \infty )$ between the empirical measure of independent and identically distributed ${\mathbb R}^d$-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order $rp$ for some $r>1$, and we discuss the optimality of the bounds.
</p>projecteuclid.org/euclid.ejp/1550113245_20190716040113Tue, 16 Jul 2019 04:01 EDTProfile of a self-similar growth-fragmentationhttps://projecteuclid.org/euclid.ejp/1550199785<strong>François Gaston Ged</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
A self-similar growth-fragmentation describes the evolution of particles that grow and split as time passes. Its genealogy yields a self-similar continuum tree endowed with an intrinsic measure. Extending results of Haas [13] for pure fragmentations, we relate the existence of an absolutely continuous profile to a simple condition in terms of the index of self-similarity and the so-called cumulant of the growth-fragmentation. When absolutely continuous, we approximate the profile by a function of the small fragments, and compute the Hausdorff dimension in the singular case. Applications to Boltzmann random planar maps are emphasized, exploiting recently established connections between growth-fragmentations and random maps [1, 5].
</p>projecteuclid.org/euclid.ejp/1550199785_20190716040113Tue, 16 Jul 2019 04:01 EDTSpectral conditions for equivalence of Gaussian random fields with stationary incrementshttps://projecteuclid.org/euclid.ejp/1550199786<strong>Abolfazl Safikhani</strong>, <strong>Yimin Xiao</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 19 pp..</p><p><strong>Abstract:</strong><br/>
This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.
</p>projecteuclid.org/euclid.ejp/1550199786_20190716040113Tue, 16 Jul 2019 04:01 EDTUniversality of the least singular value for sparse random matriceshttps://projecteuclid.org/euclid.ejp/1550221265<strong>Ziliang Che</strong>, <strong>Patrick Lopatto</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 53 pp..</p><p><strong>Abstract:</strong><br/>
We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution with parameter $p$. These matrices represent the adjacency matrices of random Erdős–Rényi digraphs and are sparse when $p\ll 1$. We prove that in the regime $pN\gg 1$, the distribution of the least singular value is universal in the sense that it is independent of $p$ and equal to the distribution of the least singular value of a Gaussian matrix ensemble. We also prove the universality of the joint distribution of multiple small singular values. Our methods extend to matrix ensembles whose entries are chosen from arbitrary distributions that may be correlated, complex valued, and have unequal variances.
</p>projecteuclid.org/euclid.ejp/1550221265_20190716040113Tue, 16 Jul 2019 04:01 EDTConvergence of the empirical spectral distribution of Gaussian matrix-valued processeshttps://projecteuclid.org/euclid.ejp/1550286034<strong>Arturo Jaramillo</strong>, <strong>Juan Carlos Pardo</strong>, <strong>José Luis Pérez</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 22 pp..</p><p><strong>Abstract:</strong><br/>
For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(\lambda _{1}^{(n)}(t),\dots , \lambda _{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $\mu ^{(n)}=(\mu _{t}^{(n)}; t\geq 0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $\mu ^{(n)}$ converges in probability to a deterministic limit $\mu $, in the topology of uniform convergence over compact sets. We show that the process $\mu $ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers’ equation. Our results extend those of Rogers and Shi [14] for the free Brownian motion and Pardo et al. [12] for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.
</p>projecteuclid.org/euclid.ejp/1550286034_20190716040113Tue, 16 Jul 2019 04:01 EDTSmall-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak Hörmander typehttps://projecteuclid.org/euclid.ejp/1550480425<strong>Karen Habermann</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 19 pp..</p><p><strong>Abstract:</strong><br/>
We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak Hörmander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expression for the bridge from $0$ to $0$ in time $1$ of an iterated Kolmogorov diffusion.
</p>projecteuclid.org/euclid.ejp/1550480425_20190716040113Tue, 16 Jul 2019 04:01 EDTNon-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficienthttps://projecteuclid.org/euclid.ejp/1550653271<strong>Benjamin Jourdain</strong>, <strong>Ahmed Kebaier</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 34 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.
</p>projecteuclid.org/euclid.ejp/1550653271_20190716040113Tue, 16 Jul 2019 04:01 EDTNon asymptotic variance bounds and deviation inequalities by optimal transporthttps://projecteuclid.org/euclid.ejp/1550653272<strong>Kevin Tanguy</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 18 pp..</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\mathbb{R} ^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.
</p>projecteuclid.org/euclid.ejp/1550653272_20190716040113Tue, 16 Jul 2019 04:01 EDTA note on concentration for polynomials in the Ising modelhttps://projecteuclid.org/euclid.ejp/1555466612<strong>Radosław Adamczak</strong>, <strong>Michał Kotowski</strong>, <strong>Bartłomiej Polaczyk</strong>, <strong>Michał Strzelecki</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 22 pp..</p><p><strong>Abstract:</strong><br/>
We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson–Wright type inequality.
We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.
</p>projecteuclid.org/euclid.ejp/1555466612_20190716040113Tue, 16 Jul 2019 04:01 EDTAsymptotic representation theory and the spectrum of a random geometric graph on a compact Lie grouphttps://projecteuclid.org/euclid.ejp/1555466613<strong>Pierre-Loïc Méliot</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 85 pp..</p><p><strong>Abstract:</strong><br/>
Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $\Gamma _{\mathrm{geom} }(N,L)$ whose vertices are $N$ random points $g_1,\ldots ,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $\Gamma _{\mathrm{geom} }(N,L)$, when $N$ goes to infinity.
1. If $L$ is fixed and $N \to + \infty $ (Gaussian regime), then the largest eigenvalues of $\Gamma _{\mathrm{geom} }(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions.
2. If $L = O(N^{-\frac{1} {\dim G}})$ and $N \to +\infty $ (Poissonian regime), then the geometric graph $\Gamma _{\mathrm{geom} }(N,L)$ converges in the local Benjamini–Schramm sense, which implies the weak convergence in probability of the spectral measure of $\Gamma _{\mathrm{geom} }(N,L)$.
In both situations, the representation theory of the group $G$ provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl’s character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of $G$.
</p>projecteuclid.org/euclid.ejp/1555466613_20190716040113Tue, 16 Jul 2019 04:01 EDTEdge universality of correlated Gaussianshttps://projecteuclid.org/euclid.ejp/1556179228<strong>Arka Adhikari</strong>, <strong>Ziliang Che</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 25 pp..</p><p><strong>Abstract:</strong><br/>
We consider a Gaussian random matrix with correlated entries that have a power law decay of order $d>2$ and prove universality for the extreme eigenvalues. A local law is proved using the self-consistent equation combined with a decomposition of the matrix. This local law along with concentration of eigenvalues around the edge allows us to get a bound for extreme eigenvalues. Using a recent result of the Dyson-Brownian motion, we prove universality of extreme eigenvalues.
</p>projecteuclid.org/euclid.ejp/1556179228_20190716040113Tue, 16 Jul 2019 04:01 EDTAnalysis of large urn models with local mean-field interactionshttps://projecteuclid.org/euclid.ejp/1557453644<strong>Wen Sun</strong>, <strong>Robert Philippe</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 33 pp..</p><p><strong>Abstract:</strong><br/>
The stochastic models investigated in this paper describe the evolution of a set of $F_N$ identical balls scattered into $N$ urns connected by an underlying symmetrical graph with constant degree $h_N$. After some random amount of time all the balls of any urn are redistributed locally, among the $h_N$ urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality $h_N$ of the range of interaction is not necessarily linear with respect to $N$ as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns located in the neighborhood of a given urn. Under convenient conditions, by taking an appropriate Wasserstein distance and by establishing several technical estimates for local empirical distributions, we are able to prove mean-field convergence results.
When the load per node goes to infinity, a convergence result for the invariant distribution of the associated McKean-Vlasov process is obtained for several allocation policies. For the class of power of $d$ choices policies, we show that the associated invariant measure has an asymptotic finite support property under this regime. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of $d$ choices policies.
</p>projecteuclid.org/euclid.ejp/1557453644_20190716040113Tue, 16 Jul 2019 04:01 EDTStrong renewal theorems and local large deviations for multivariate random walks and renewalshttps://projecteuclid.org/euclid.ejp/1557453645<strong>Quentin Berger</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 47 pp..</p><p><strong>Abstract:</strong><br/>
We study a random walk $\mathbf{S} _n$ on $\mathbb{Z} ^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha } =(\alpha _1,\ldots ,\alpha _d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, i.e. a sharp asymptotic of the Green function $G(\mathbf{0} ,\mathbf{x} )$ as $\|\mathbf{x} \|\to +\infty $, along the “favorite direction or scaling”: (i) if $\sum _{i=1}^d \alpha _i^{-1} < 2$ (reminiscent of Garsia-Lamperti’s condition when $d=1$ [17]); (ii) if a certain local condition holds (reminiscent of Doney’s [13, Eq. (1.9)] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0} ,\mathbf{x} )$, sharpening estimates when $\mathbf{x} $ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha _i\equiv \alpha $, in the favorite scaling, and has even left aside the case $\alpha \in [1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.
</p>projecteuclid.org/euclid.ejp/1557453645_20190716040113Tue, 16 Jul 2019 04:01 EDTHeight and contour processes of Crump-Mode-Jagers forests (II): the Bellman–Harris universality classhttps://projecteuclid.org/euclid.ejp/1558145015<strong>Emmanuel Schertzer</strong>, <strong>Florian Simatos</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 38 pp..</p><p><strong>Abstract:</strong><br/>
Crump–Mode–Jagers (CMJ) trees generalize Galton–Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we exhibit a simple condition under which the height and contour processes of CMJ forests belong to the universality class of Bellman–Harris processes. This condition formalizes an asymptotic independence between the chronological and genealogical structures. We show that it is satisfied by a large class of CMJ processes and in particular, quite surprisingly, by CMJ processes with a finite variance offspring distribution. Along the way, we prove a general tightness result.
</p>projecteuclid.org/euclid.ejp/1558145015_20190716040113Tue, 16 Jul 2019 04:01 EDTInvariance principle for non-homogeneous random walkshttps://projecteuclid.org/euclid.ejp/1558145016<strong>Nicholas Georgiou</strong>, <strong>Aleksandar Mijatović</strong>, <strong>Andrew R. Wade</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 38 pp..</p><p><strong>Abstract:</strong><br/>
We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in ${\mathbb R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X} $ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq 2$. To characterize $\mathcal{X} $, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in ${\mathbb R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X} $ and thus develop the excursion theory of $\mathcal{X} $ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X} $ in ${\mathbb R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X} $ is time-reversible. If so, the excursions of $\mathcal{X} $ in ${\mathbb R}^d$ generalize the classical Pitman–Yor splitting-at-the-maximum property of Bessel excursions.
</p>projecteuclid.org/euclid.ejp/1558145016_20190716040113Tue, 16 Jul 2019 04:01 EDTOn self-avoiding polygons and walks: the snake method via polygon joininghttps://projecteuclid.org/euclid.ejp/1558404407<strong>Alan Hammond</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 43 pp..</p><p><strong>Abstract:</strong><br/>
For $d \geq 2$ and $n \in \mathbb{N} $, let $\mathsf{W} _n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\mathbb{Z} ^d$, and write $\Gamma $ for a $\mathsf{W} _n$-distributed walk. We show that the closing probability $\mathsf{W} _n \big (\vert \vert \Gamma _n \vert \vert = 1 \big )$ that $\Gamma $’s endpoint neighbours the origin is at most $n^{-4/7 + o(1)}$ for a positive density set of odd $n$ in dimension $d = 2$. This result is proved using the snake method, a general technique for proving closing probability upper bounds, which originated in [3] and was made explicit in [8]. Our conclusion is reached by applying the snake method in unison with a polygon joining technique whose use was initiated by Madras in [13].
</p>projecteuclid.org/euclid.ejp/1558404407_20190716040113Tue, 16 Jul 2019 04:01 EDTFree energy of directed polymers in random environment in $1+1$-dimension at high temperaturehttps://projecteuclid.org/euclid.ejp/1558404408<strong>Makoto Nakashima</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 43 pp..</p><p><strong>Abstract:</strong><br/>
We consider the free energy $F(\beta )$ of the directed polymers in random environment in $1+1$-dimension. It is known that $F(\beta )$ is of order $-\beta ^4$ as $\beta \to 0$ [3, 28, 42]. In this paper, we will prove that under a certain dimension free concentration condition on the potential, \[ \lim _{\beta \to 0}\frac{F(\beta )} {\beta ^4}=\lim _{T\to \infty }\frac{1} {T}P_\mathcal{Z} \left [\log \mathcal{Z} _{\sqrt{2} }(T)\right ] =-\frac{1} {6}, \] where $\{\mathcal{Z} _\beta (t,x):t\geq 0,x\in \mathbb{R} \}$ is the unique mild solution to the stochastic heat equation \[ \frac{\partial } {\partial t}\mathcal{Z} =\frac{1} {2}\Delta \mathcal{Z} +\beta \mathcal{Z} {\dot{\mathcal W} },\ \ \lim _{t\to 0}\mathcal{Z} (t,x)dx=\delta _{0}(dx), \] where $\mathcal{W} $ is a time-space white noise and \[ \mathcal{Z} _\beta (t)=\int _\mathbb{R} \mathcal{Z} _\beta (t,x)dx. \]
</p>projecteuclid.org/euclid.ejp/1558404408_20190716040113Tue, 16 Jul 2019 04:01 EDTFrom the master equation to mean field game limit theory: a central limit theoremhttps://projecteuclid.org/euclid.ejp/1558576902<strong>François Delarue</strong>, <strong>Daniel Lacker</strong>, <strong>Kavita Ramanan</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 54 pp..</p><p><strong>Abstract:</strong><br/>
Mean field games (MFGs) describe the limit, as $n$ tends to infinity, of stochastic differential games with $n$ players interacting with one another through their common empirical distribution. Under suitable smoothness assumptions that guarantee uniqueness of the MFG equilibrium, a form of law of large of numbers (LLN), also known as propagation of chaos, has been established to show that the MFG equilibrium arises as the limit of the sequence of empirical measures of the $n$-player game Nash equilibria, including the case when player dynamics are driven by both idiosyncratic and common sources of noise. The proof of convergence relies on the so-called master equation for the value function of the MFG, a partial differential equation on the space of probability measures. In this work, under additional assumptions, we establish a functional central limit theorem (CLT) that characterizes the limiting fluctuations around the LLN limit as the unique solution of a linear stochastic PDE. The key idea is to use the solution to the master equation to construct an associated McKean-Vlasov interacting $n$-particle system that is sufficiently close to the Nash equilibrium dynamics of the $n$-player game for large $n$. We then derive the CLT for the latter from the CLT for the former. Along the way, we obtain a new multidimensional CLT for McKean-Vlasov systems. We also illustrate the broader applicability of our methodology by applying it to establish a CLT for a specific linear-quadratic example that does not satisfy our main assumptions, and we explicitly solve the resulting stochastic PDE in this case.
</p>projecteuclid.org/euclid.ejp/1558576902_20190716040113Tue, 16 Jul 2019 04:01 EDTBranching trees I: concatenation and infinite divisibilityhttps://projecteuclid.org/euclid.ejp/1559354444<strong>Patric Glöde</strong>, <strong>Andreas Greven</strong>, <strong>Thomas Rippl</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 55 pp..</p><p><strong>Abstract:</strong><br/>
The goal of this work is to decompose random populations with a genealogy in subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible genealogies. We model the genealogical structure of a population by (equivalence classes of) ultrametric measure spaces (um-spaces) as elements of the Polish space $\mathbb{U} $ which we recall. In order to then analyze the family structure in this coding we introduce an algebraic structure on um-spaces (a consistent collection of semigroups). This allows us to obtain a path of decompositions of subfamilies of fixed kinship $ h $ (described as ultrametric measure spaces), for every depth $ h $ as a measurable functional of the genealogy.
Technically the elements of the semigroup are those um-spaces which have diameter less or equal to $2h$ called $h$ -forests ($h> 0$). They arise from a given ultrametric measure space by applying maps called $ h-$truncation. We can define a concatenation of two $ h$-forests as binary operation. The corresponding semigroup is a Delphic semigroup and any $h$-forest has a unique prime factorization in $h$-trees (um-spaces of diameter less than $2h$). Therefore we have a nested $\mathbb{R} ^{+}$-indexed consistent (they arise successively by truncation) collection of Delphic semigroups with unique prime factorization.
Random elements in the semigroup are studied, in particular infinitely divisible random variables. Here we define infinite divisibility of random genealogies as the property that the $h$-tops can be represented as concatenation of independent identically distributed h-forests for every $h$ and obtain a Lévy-Khintchine representation of this object and a corresponding representation via a concatenation of points of a Poisson point process of h-forests.
Finally the case of discrete and marked um-spaces is treated allowing to apply the results to both the individual based and most important spatial populations.
The results have various applications. In particular the case of the genealogical ($\mathbb{U} $-valued) Feller diffusion and genealogical ($\mathbb{U} ^{V}$-valued) super random walk is treated based on the present work in [13] and [24].
In the part II of this paper we go in a different direction and refine the study in the case of continuum branching populations, give a refined analysis of the Laplace functional and give a representation in terms of a Cox process on h-trees, rather than forests.
</p>projecteuclid.org/euclid.ejp/1559354444_20190716040113Tue, 16 Jul 2019 04:01 EDTStopping with expectation constraints: 3 points sufficehttps://projecteuclid.org/euclid.ejp/1561687599<strong>Stefan Ankirchner</strong>, <strong>Nabil Kazi-Tani</strong>, <strong>Maike Klein</strong>, <strong>Thomas Kruse</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 16 pp..</p><p><strong>Abstract:</strong><br/>
We consider the problem of optimally stopping a one-dimensional regular continuous strong Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The proof uses recent results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures.
</p>projecteuclid.org/euclid.ejp/1561687599_20190716040113Tue, 16 Jul 2019 04:01 EDTQuickSort: improved right-tail asymptotics for the limiting distribution, and large deviationshttps://projecteuclid.org/euclid.ejp/1561687600<strong>James Allen Fill</strong>, <strong>Wei-Chun Hung</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 13 pp..</p><p><strong>Abstract:</strong><br/>
We substantially refine asymptotic logarithmic upper bounds produced by Svante Janson (2015) on the right tail of the limiting QuickSort distribution function $F$ and by Fill and Hung (2018) on the right tails of the corresponding density $f$ and of the absolute derivatives of $f$ of each order. For example, we establish an upper bound on $\log [1 - F(x)]$ that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order $(\log x)^{2}$; the corresponding order for the Janson (2015) bound is the lead order, $x \log x$.
Using the refined asymptotic bounds on $F$, we derive right-tail large deviation (LD) results for the distribution of the number of comparisons required by QuickSort that substantially sharpen the two-sided LD results of McDiarmid and Hayward (1996).
</p>projecteuclid.org/euclid.ejp/1561687600_20190716040113Tue, 16 Jul 2019 04:01 EDTShape theorem and surface fluctuation for Poisson cylindershttps://projecteuclid.org/euclid.ejp/1561687601<strong>Marcelo Hilario</strong>, <strong>Xinyi Li</strong>, <strong>Petr Panov</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 16 pp..</p><p><strong>Abstract:</strong><br/>
We prove a shape theorem for Poisson cylinders, and give a power-law bound on surface fluctuations. In particular, we show that for any $a \in (1/2, 1)$, conditioned on the origin being in the set of cylinders, if a point belongs to this set and has Euclidean norm below $R$, then this point lies at internal distance less than $R + O(R^{a})$ from the origin.
</p>projecteuclid.org/euclid.ejp/1561687601_20190716040113Tue, 16 Jul 2019 04:01 EDTRandom walks in a moderately sparse random environmenthttps://projecteuclid.org/euclid.ejp/1561687602<strong>Dariusz Buraczewski</strong>, <strong>Piotr Dyszewski</strong>, <strong>Alexander Iksanov</strong>, <strong>Alexander Marynych</strong>, <strong>Alexander Roitershtein</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 44 pp..</p><p><strong>Abstract:</strong><br/>
A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_{n})_{n\in \mathbb{N} \cup \{0\}}$ in a sparse random environment $(S_{k},\lambda _{k})_{k\in \mathbb{Z} }$ is a nearest neighbor random walk on $\mathbb{Z} $ that jumps to the left or to the right with probability $1/2$ from every point of $\mathbb{Z} \setminus \{\ldots ,S_{-1},S_{0}=0,S_{1},\ldots \}$ and jumps to the right (left) with the random probability $\lambda _{k+1}$ ($1-\lambda _{k+1}$) from the point $S_{k}$, $k\in \mathbb{Z} $. Assuming that $(S_{k}-S_{k-1},\lambda _{k})_{k\in \mathbb{Z} }$ are independent copies of a random vector $(\xi ,\lambda )\in \mathbb{N} \times (0,1)$ and the mean $\mathbb{E} \xi $ is finite (moderate sparsity) we obtain stable limit laws for $X_{n}$, properly normalized and centered, as $n\to \infty $. While the case $\xi \leq M$ a.s. for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $\mathbb{E} \xi =\infty $ (strong sparsity) will be analyzed in a forthcoming paper.
</p>projecteuclid.org/euclid.ejp/1561687602_20190716040113Tue, 16 Jul 2019 04:01 EDTInverting the coupling of the signed Gaussian free field with a loop-souphttps://projecteuclid.org/euclid.ejp/1561687603<strong>Titus Lupu</strong>, <strong>Christophe Sabot</strong>, <strong>Pierre Tarrès</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 28 pp..</p><p><strong>Abstract:</strong><br/>
Lupu introduced a coupling between a random walk loop-soup and a Gaussian free field, where the sign of the field is constant on each cluster of loops. This coupling is a signed version of isomorphism theorems relating the square of the GFF to the occupation field of Markovian trajectories. His construction starts with a loop-soup, and by adding additional randomness samples a GFF out of it. In this article we provide the inverse construction: starting from a signed free field and using a self-interacting random walk related to this field, we construct a random walk loop-soup. Our construction relies on the previous work by Sabot and Tarrès, which inverts the coupling from the square of the GFF rather than the signed GFF itself.
</p>projecteuclid.org/euclid.ejp/1561687603_20190716040113Tue, 16 Jul 2019 04:01 EDTPhase singularities in complex arithmetic random waveshttps://projecteuclid.org/euclid.ejp/1561687604<strong>Federico Dalmao</strong>, <strong>Ivan Nourdin</strong>, <strong>Giovanni Peccati</strong>, <strong>Maurizia Rossi</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 45 pp..</p><p><strong>Abstract:</strong><br/>
Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-Itô chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations. As a by-product of our analysis, we also deduce explicit bounds in smooth distances for the second order non-central results evoked above.
</p>projecteuclid.org/euclid.ejp/1561687604_20190716040113Tue, 16 Jul 2019 04:01 EDTLocal large deviations and the strong renewal theoremhttps://projecteuclid.org/euclid.ejp/1561687605<strong>Francesco Caravenna</strong>, <strong>Ron Doney</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 48 pp..</p><p><strong>Abstract:</strong><br/>
We establish two different, but related results for random walks in the domain of attraction of a stable law of index $\alpha $. The first result is a local large deviation upper bound, valid for $\alpha \in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $\alpha \in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long-standing problem, which dates back to the 1962 paper of Garsia and Lamperti [GL62] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Wil68] for general random walks.
</p>projecteuclid.org/euclid.ejp/1561687605_20190716040113Tue, 16 Jul 2019 04:01 EDTMixing times for exclusion processes on hypergraphshttps://projecteuclid.org/euclid.ejp/1561687606<strong>Stephen B. Connor</strong>, <strong>Richard J. Pymar</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 48 pp..</p><p><strong>Abstract:</strong><br/>
We introduce a natural extension of the exclusion process to hypergraphs and prove an upper bound for its mixing time. In particular we show the existence of a constant $C$ such that for any connected, regular hypergraph $G$ within some natural class, the $\varepsilon $-mixing time of the exclusion process on $G$ with any feasible number of particles can be upper-bounded by $CT_{\text{EX} (2,G)}\log (|V|/\varepsilon )$, where $|V|$ is the number of vertices in $G$ and $T_{\text{EX} (2,G)}$ is the 1/4-mixing time of the corresponding exclusion process with just two particles. Moreover we show this is optimal in the sense that there exist hypergraphs in the same class for which $T_{\mathrm{EX} (2,G)}$ and the mixing time of just one particle are not comparable. The proofs involve an adaptation of the chameleon process , a technical tool invented by Morris ([14]) and developed by Oliveira ([15]) for studying the exclusion process on a graph.
</p>projecteuclid.org/euclid.ejp/1561687606_20190716040113Tue, 16 Jul 2019 04:01 EDTUniqueness and non-uniqueness for spin-glass ground states on treeshttps://projecteuclid.org/euclid.ejp/1563264040<strong>Johannes Bäumler</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 17 pp..</p><p><strong>Abstract:</strong><br/>
We consider a spin glass at temperature $T = 0$ where the underlying graph is a locally finite tree. We prove for a wide range of coupling distributions that uniqueness of ground states is equivalent to the maximal flow from any vertex to $\infty $ (where each edge $e$ has capacity $|J_{e}|$) being equal to zero which is equivalent to recurrence of the simple random walk on the tree.
</p>projecteuclid.org/euclid.ejp/1563264040_20190716040113Tue, 16 Jul 2019 04:01 EDTFront evolution of the Fredrickson-Andersen one spin facilitated modelhttps://projecteuclid.org/euclid.ejp/1546571126<strong>Oriane Blondel</strong>, <strong>Aurelia Deshayes</strong>, <strong>Cristina Toninelli</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 32 pp..</p><p><strong>Abstract:</strong><br/>
The Fredrickson-Andersen one spin facilitated model (FA-1f) on $\mathbb Z$ belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability $q$ (respectively $p=1-q$), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a conﬁguration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for $q$ larger than a threshold $\bar q<1$, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.
</p>projecteuclid.org/euclid.ejp/1546571126_20190805220401Mon, 05 Aug 2019 22:04 EDTHeavy subtrees of Galton-Watson trees with an application to Apollonian networkshttps://projecteuclid.org/euclid.ejp/1549357219<strong>Luc Devroye</strong>, <strong>Cecilia Holmgren</strong>, <strong>Henning Sulzbach</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 44 pp..</p><p><strong>Abstract:</strong><br/>
We study heavy subtrees of conditional Galton-Watson trees. In a standard Galton-Watson tree conditional on its size being $n$, we order all children by their subtree sizes, from large (heavy) to small. A node is marked if it is among the $k$ heaviest nodes among its siblings. Unmarked nodes and their subtrees are removed, leaving only a tree of marked nodes, which we call the $k$-heavy tree. We study various properties of these trees, including their size and the maximal distance from any original node to the $k$-heavy tree. In particular, under some moment condition, the $2$-heavy tree is with high probability larger than $cn$ for some constant $c > 0$, and the maximal distance from the $k$-heavy tree is $O(n^{1/(k+1)})$ in probability. As a consequence, for uniformly random Apollonian networks of size $n$, the expected size of the longest simple path is $\Omega (n)$. We also show that the length of the heavy path (that is, $k=1$) converges (after rescaling) to the corresponding object in Aldous’ Brownian continuum random tree.
</p>projecteuclid.org/euclid.ejp/1549357219_20190805220401Mon, 05 Aug 2019 22:04 EDTDifferentiability of SDEs with drifts of super-linear growthhttps://projecteuclid.org/euclid.ejp/1549616424<strong>Peter Imkeller</strong>, <strong>Gonçalo dos Reis</strong>, <strong>William Salkeld</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 43 pp..</p><p><strong>Abstract:</strong><br/>
We close an unexpected gap in the literature of Stochastic Differential Equations (SDEs) with drifts of super linear growth and with random coefficients, namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Stochastic Gâteaux Differentiability and Ray Absolute Continuity. This method enables one to take limits in probability rather than mean square or almost surely bypassing the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology of [13, Lemma 1.2.3] for this setting. Several examples illustrating the range and scope of our results are presented.
We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.
</p>projecteuclid.org/euclid.ejp/1549616424_20190805220401Mon, 05 Aug 2019 22:04 EDTA stability approach for solving multidimensional quadratic BSDEshttps://projecteuclid.org/euclid.ejp/1549616425<strong>Jonathan Harter</strong>, <strong>Adrien Richou</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 51 pp..</p><p><strong>Abstract:</strong><br/>
We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDE). This class is characterized by constraints on some uniform a priori estimate on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921–2939] concerning scalar quadratic BSDEs.
</p>projecteuclid.org/euclid.ejp/1549616425_20190805220401Mon, 05 Aug 2019 22:04 EDTCan the stochastic wave equation with strong drift hit zero?https://projecteuclid.org/euclid.ejp/1550653273<strong>Kevin Lin</strong>, <strong>Carl Mueller</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 26 pp..</p><p><strong>Abstract:</strong><br/>
We study the stochastic wave equation with multiplicative noise and singular drift:
\[\partial _{t}u(t,x)=\Delta u(t,x)+u^{-\alpha }(t,x)+g(u(t,x))\dot{W} (t,x)\]
where $x$ lies in the circle $\mathbf{R} /J\mathbf{Z} $ and $u(0,x)>0$. We show that
(i) If $0<\alpha <1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$.
(ii) If $\alpha >3$ then with probability one, $u(t,x)\ne 0$ for all $(t,x)$.
</p>projecteuclid.org/euclid.ejp/1550653273_20190805220401Mon, 05 Aug 2019 22:04 EDTAsymptotic properties of expansive Galton-Watson treeshttps://projecteuclid.org/euclid.ejp/1550826098<strong>Romain Abraham</strong>, <strong>Jean-François Delmas</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 51 pp..</p><p><strong>Abstract:</strong><br/>
We consider a super-critical Galton-Watson tree $\tau $ whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau _n$ distributed as $\tau $ conditioned on the $n$-th generation, $Z_n$, to be of size $a_n\in{\mathbb N} $. We identify the possible local limits of $\tau _n$ as $n$ goes to infinity according to the growth rate of $a_n$. In the low regime, the local limit $\tau ^0$ is the Kesten tree, in the moderate regime the family of local limits, $\tau ^\theta $ for $\theta \in (0, +\infty )$, is distributed as $\tau $ conditionally on $\{W=\theta \}$, where $W$ is the (non-trivial) limit of the renormalization of $Z_n$. In the high regime, we prove the local convergence towards $\tau ^\infty $ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits $(\tau ^\theta , \theta \in [0, \infty ])$.
</p>projecteuclid.org/euclid.ejp/1550826098_20190805220401Mon, 05 Aug 2019 22:04 EDTExceedingly large deviations of the totally asymmetric exclusion processhttps://projecteuclid.org/euclid.ejp/1550826099<strong>Stefano Olla</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 71 pp..</p><p><strong>Abstract:</strong><br/>
Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h} (t,x) $ under the hyperbolic scaling of space and time by $ N $, i.e., $ \mathsf{h} _{N}(t,\xi ) := \frac{1} {N}\mathsf{h} (Nt,N\xi ) $. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $ \exp (-O(N)) $, referred to as speed-$ N $; while the other with probability $ \exp (-O(N^{2})) $, referred to as speed-$ N^2 $. In this work we study the speed-$ N^2 $ functional Large Deviation Principle (LDP) of the TASEP, and establish (non-matching) large deviation upper and lower bounds.
</p>projecteuclid.org/euclid.ejp/1550826099_20190805220401Mon, 05 Aug 2019 22:04 EDTCramér’s estimate for stable processes with power drifthttps://projecteuclid.org/euclid.ejp/1551150461<strong>Christophe Profeta</strong>, <strong>Thomas Simon</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.
</p>projecteuclid.org/euclid.ejp/1551150461_20190805220401Mon, 05 Aug 2019 22:04 EDTFinding the seed of uniform attachment treeshttps://projecteuclid.org/euclid.ejp/1551323285<strong>Gábor Lugosi</strong>, <strong>Alan S. Pereira</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 15 pp..</p><p><strong>Abstract:</strong><br/>
A uniform attachment tree is a random tree that is generated dynamically. Starting from a fixed “seed” tree, vertices are added sequentially by attaching each vertex to an existing vertex chosen uniformly at random. Upon observing a large (unlabeled) tree, one wishes to find the initial seed. We investigate to what extent seed trees can be recovered, at least partially. We consider three types of seeds: a path, a star, and a random uniform attachment tree. We propose and analyze seed-finding algorithms for all three types of seed trees.
</p>projecteuclid.org/euclid.ejp/1551323285_20190805220401Mon, 05 Aug 2019 22:04 EDTScaling limits of population and evolution processes in random environmenthttps://projecteuclid.org/euclid.ejp/1552013626<strong>Vincent Bansaye</strong>, <strong>Maria-Emilia Caballero</strong>, <strong>Sylvie Méléard</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 38 pp..</p><p><strong>Abstract:</strong><br/>
We propose a general method for investigating scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively defined as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.
</p>projecteuclid.org/euclid.ejp/1552013626_20190805220401Mon, 05 Aug 2019 22:04 EDTDirected, cylindric and radial Brownian webshttps://projecteuclid.org/euclid.ejp/1553133829<strong>David Coupier</strong>, <strong>Jean-François Marckert</strong>, <strong>Viet Chi Tran</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 48 pp..</p><p><strong>Abstract:</strong><br/>
The Brownian web (BW) is a collection of coalescing Brownian paths $(W_{(x,t)},(x,t) \in \mathbb{R} ^2)$ indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in Coletti and Valencia is shown to converge to the CBW.
</p>projecteuclid.org/euclid.ejp/1553133829_20190805220401Mon, 05 Aug 2019 22:04 EDTGlobal fluctuations for 1D log-gas dynamics. Covariance kernel and supporthttps://projecteuclid.org/euclid.ejp/1553155301<strong>Jeremie Unterberger</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 28 pp..</p><p><strong>Abstract:</strong><br/>
We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations,
\[ d\lambda _t^i=\frac{1} {\sqrt{N} } dW_t^i - V'(\lambda _t^i) dt+ \frac{\beta } {2N} \sum _{j\not =i} \frac{dt} {\lambda ^i_t-\lambda ^j_t}, \qquad i=1,\ldots ,N, \qquad \mbox{(0.1)} \]
with $\beta >1$, sometimes called generalized Dyson’s Brownian motion , describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta $-ensemble, with sufficiently regular convex potential $V$. The limit $N\to \infty $ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown [39] to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation.
We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $\rho _t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.
</p>projecteuclid.org/euclid.ejp/1553155301_20190805220401Mon, 05 Aug 2019 22:04 EDTMixing times for the simple exclusion process in ballistic random environmenthttps://projecteuclid.org/euclid.ejp/1553155302<strong>Dominik Schmid</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 25 pp..</p><p><strong>Abstract:</strong><br/>
We consider the exclusion process on segments of the integers in a site-dependent random environment. We assume to be in the ballistic regime in which a single particle has positive linear speed. Our goal is to study the mixing time of the exclusion process when the number of particles is linear in the size of the segment. We investigate the order of the mixing time depending on the support of the environment distribution. In particular, we prove for nestling environments that the order of the mixing time is different than in the case of a single particle.
</p>projecteuclid.org/euclid.ejp/1553155302_20190805220401Mon, 05 Aug 2019 22:04 EDTThe speed of critically biased random walk in a one-dimensional percolation modelhttps://projecteuclid.org/euclid.ejp/1553306439<strong>Jan-Erik Lübbers</strong>, <strong>Matthias Meiners</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 29 pp..</p><p><strong>Abstract:</strong><br/>
We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and Häggström and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z} ^d$, namely, for some critical value $\lambda _{\mathrm{c} }>0$ of the bias, it holds that the asymptotic linear speed $\overline{\mathrm {v}} $ of the walk is strictly positive if the bias $\lambda $ is strictly smaller than $\lambda _{\mathrm{c} }$, whereas $\overline{\mathrm {v}} =0$ if $\lambda \geq \lambda _{\mathrm{c} }$.
We show that at the critical bias $\lambda = \lambda _{\mathrm{c} }$, the displacement of the random walk from the origin is of order $n/\log n$. This is in accordance with simulation results by Dhar and Stauffer for biased random walk on the infinite cluster of supercritical bond percolation on $\mathbb{Z} ^d$.
Our result is based on fine estimates for the tails of suitable regeneration times. As a by-product of these estimates we also obtain the order of fluctuations of the walk in the sub-ballistic and in the ballistic, nondiffusive phase.
</p>projecteuclid.org/euclid.ejp/1553306439_20190805220401Mon, 05 Aug 2019 22:04 EDTSplitting tessellations in spherical spaceshttps://projecteuclid.org/euclid.ejp/1553565775<strong>Daniel Hug</strong>, <strong>Christoph Thäle</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 60 pp..</p><p><strong>Abstract:</strong><br/>
The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $d\geq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $k\in \{1,\ldots ,d-1\}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.
</p>projecteuclid.org/euclid.ejp/1553565775_20190805220401Mon, 05 Aug 2019 22:04 EDTSecond Errata to “Processes on Unimodular Random Networks”https://projecteuclid.org/euclid.ejp/1553565776<strong>David Aldous</strong>, <strong>Russell Lyons</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 2 pp..</p><p><strong>Abstract:</strong><br/>
We correct a few more minor errors in our paper, Electron. J. Probab. 12 , Paper 54 (2007), 1454–1508.
</p>projecteuclid.org/euclid.ejp/1553565776_20190805220401Mon, 05 Aug 2019 22:04 EDTQuantitative contraction rates for Markov chains on general state spaceshttps://projecteuclid.org/euclid.ejp/1553565777<strong>Andreas Eberle</strong>, <strong>Mateusz B. Majka</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 36 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich ($L^1$ Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on $\mathbb R^d$ with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.
</p>projecteuclid.org/euclid.ejp/1553565777_20190805220401Mon, 05 Aug 2019 22:04 EDTMultivariate approximation in total variation using local dependencehttps://projecteuclid.org/euclid.ejp/1553565778<strong>A.D. Barbour</strong>, <strong>A. Xia</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 35 pp..</p><p><strong>Abstract:</strong><br/>
We establish two theorems for assessing the accuracy in total variation of multivariate discrete normal approximation to the distribution of an integer valued random vector $W$. The first is for sums of random vectors whose dependence structure is local. The second applies to random vectors $W$ resulting from integrating the ${\mathbb Z}^d$-valued marks of a marked point process with respect to its ground process. The error bounds are of magnitude comparable to those given in [Rinott & Rotar (1996)], but now with respect to the stronger total variation distance. Instead of requiring the summands to be bounded, we make third moment assumptions. We demonstrate the use of the theorems in four applications: monochrome edges in vertex coloured graphs, induced triangles and $2$-stars in random geometric graphs, the times spent in different states by an irreducible and aperiodic finite Markov chain, and the maximal points in different regions of a homogeneous Poisson point process.
</p>projecteuclid.org/euclid.ejp/1553565778_20190805220401Mon, 05 Aug 2019 22:04 EDTA random walk with catastropheshttps://projecteuclid.org/euclid.ejp/1553565779<strong>Iddo Ben-Ari</strong>, <strong>Alexander Roitershtein</strong>, <strong>Rinaldo B. Schinazi</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
Random population dynamics with catastrophes (events pertaining to possible elimination of a large portion of the population) has a long history in the mathematical literature. In this paper we study an ergodic model for random population dynamics with linear growth and binomial catastrophes: in a catastrophe, each individual survives with some fixed probability, independently of the rest. Through a coupling construction, we obtain sharp two-sided bounds for the rate of convergence to stationarity which are applied to show that the model exhibits a cutoff phenomenon.
</p>projecteuclid.org/euclid.ejp/1553565779_20190805220401Mon, 05 Aug 2019 22:04 EDTOn Stein’s method for multivariate self-decomposable laws with finite first momenthttps://projecteuclid.org/euclid.ejp/1553565780<strong>Benjamin Arras</strong>, <strong>Christian Houdré</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 33 pp..</p><p><strong>Abstract:</strong><br/>
We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R} ^d$ having ﬁnite ﬁrst moment. Building on previous univariate ﬁndings, we solve an integro-partial differential Stein equation by a mixture of semigroup and Fourier analytic methods. Then, under a second moment assumption, we introduce a notion of Stein kernel and an associated Stein discrepancy speciﬁcally designed for inﬁnitely divisible distributions. Combining these new tools, we obtain quantitative bounds on smooth-Wasserstein distances between a probability measure in $\mathbb{R} ^d$ and a non-degenerate self-decomposable target law with ﬁnite second moment. Finally, under an appropriate Poincaré-type inequality assumption, we investigate, via variational methods, the existence of Stein kernels. In particular, this leads to quantitative versions of classical results on characterizations of probability distributions by variational functionals.
</p>projecteuclid.org/euclid.ejp/1553565780_20190805220401Mon, 05 Aug 2019 22:04 EDTProbability measure-valued polynomial diffusionshttps://projecteuclid.org/euclid.ejp/1553565781<strong>Christa Cuchiero</strong>, <strong>Martin Larsson</strong>, <strong>Sara Svaluto-Ferro</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 32 pp..</p><p><strong>Abstract:</strong><br/>
We introduce a class of probability measure-valued diffusions, coined polynomial , of which the well-known Fleming–Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [8, 21] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.
</p>projecteuclid.org/euclid.ejp/1553565781_20190805220401Mon, 05 Aug 2019 22:04 EDTInvasion percolation on Galton-Watson treeshttps://projecteuclid.org/euclid.ejp/1554256913<strong>Marcus Michelen</strong>, <strong>Robin Pemantle</strong>, <strong>Josh Rosenberg</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 35 pp..</p><p><strong>Abstract:</strong><br/>
We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Additionally, we analyze the forward maximal weights along the backbone of the invasion cluster and prove a limit law for the process.
</p>projecteuclid.org/euclid.ejp/1554256913_20190805220401Mon, 05 Aug 2019 22:04 EDT$k$-cut on paths and some treeshttps://projecteuclid.org/euclid.ejp/1559700303<strong>Xing Shi Cai</strong>, <strong>Cecilia Holmgren</strong>, <strong>Luc Devroye</strong>, <strong>Fiona Skerman</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 22 pp..</p><p><strong>Abstract:</strong><br/>
We define the (random) $k$-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon [21] except now a node must be cut $k$ times before it is destroyed. The first order terms of the expectation and variance of $\mathcal{X} _{n}$, the $k$-cut number of a path of length $n$, are proved. We also show that $\mathcal{X} _{n}$, after rescaling, converges in distribution to a limit $\mathcal{B} _{k}$, which has a complicated representation. The paper then briefly discusses the $k$-cut number of some trees and general graphs. We conclude by some analytic results which may be of interest.
</p>projecteuclid.org/euclid.ejp/1559700303_20190805220401Mon, 05 Aug 2019 22:04 EDTA boundary local time for one-dimensional super-Brownian motion and applicationshttps://projecteuclid.org/euclid.ejp/1559700304<strong>Thomas Hughes</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 58 pp..</p><p><strong>Abstract:</strong><br/>
For a one-dimensional super-Brownian motion with density $X(t,x)$, we construct a random measure $L_{t}$ called the boundary local time which is supported on $BZ_{t} := \partial \{x:X(t,x) = 0\}$, thus confirming a conjecture of Mueller, Mytnik and Perkins [13]. $L_{t}$ is analogous to the local time at $0$ of solutions to an SDE. We establish first and second moment formulas for $L_{t}$, some basic properties, and a representation in terms of a cluster decomposition. Via the moment measures and the energy method we give a more direct proof that $\text{dim} (BZ_{t}) = 2-2\lambda _{0}> 0$ with positive probability, a recent result of Mueller, Mytnik and Perkins [13], where $-\lambda _{0}$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck operator that characterizes the left tail of $X(t,x)$. In a companion work [6], the author and Perkins use the boundary local time and some of its properties proved here to show that $\text{dim} (BZ_{t}) = 2-2\lambda _{0}$ a.s. on $\{X_{t}(\mathbb{R} ) > 0 \}$.
</p>projecteuclid.org/euclid.ejp/1559700304_20190805220401Mon, 05 Aug 2019 22:04 EDTLévy processes with finite variance conditioned to avoid an intervalhttps://projecteuclid.org/euclid.ejp/1559700305<strong>Leif Döring</strong>, <strong>Alexander R. Watson</strong>, <strong>Philip Weissmann</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 32 pp..</p><p><strong>Abstract:</strong><br/>
Conditioning Markov processes to avoid a set is a classical problem that has been studied in many settings. In the present article we study the question if a Lévy process can be conditioned to avoid an interval and, if so, the path behavior of the conditioned process. For Lévy processes with finite second moments we show that conditioning is possible and identify the conditioned process as an $h$-transform of the original killed process. The $h$-transform is explicit in terms of successive overshoot distributions and is used to prove that the conditioned process diverges to $+\infty $ and $-\infty $ with positive probabilities.
</p>projecteuclid.org/euclid.ejp/1559700305_20190805220401Mon, 05 Aug 2019 22:04 EDTAsymptotic behaviour of heavy-tailed branching processes in random environmentshttps://projecteuclid.org/euclid.ejp/1559700306<strong>Wenming Hong</strong>, <strong>Xiaoyue Zhang</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 17 pp..</p><p><strong>Abstract:</strong><br/>
Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\mathbb{E} \log m(\xi _{0})=\infty $. We show that (1) there exists no proper $c_{n}$ such that $\{Z_{n}/c_{n}\}$ has a proper, non-degenerate limit; (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\left (\bar{\xi } ,Z_{n}(\bar{\xi } )\right )$ converges almost surely to a random variable $Y(\bar{\xi } )$, where $Y\in (0,1)~\eta $-a.s.; (3) finally, we give the necessary and sufficient conditions for the almost sure convergence of $\left \{\frac{U(\bar {\xi },Z_{n}(\bar {\xi }))} {c_{n}(\bar{\xi } )}\right \}$, where $U(\bar{\xi } )$ is a slowly varying function that may depend on $\bar{\xi } $.
</p>projecteuclid.org/euclid.ejp/1559700306_20190805220401Mon, 05 Aug 2019 22:04 EDTThe stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patternshttps://projecteuclid.org/euclid.ejp/1560391565<strong>Raphaël Forien</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 35 pp..</p><p><strong>Abstract:</strong><br/>
We study a one-dimensional spatial population model where the population sizes of the subpopulations are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the effective diffusion rate of genes within the population and the effective local population density. We find that, in our model, the local heterogeneity leads to a slower effective diffusion rate and a larger effective population density than in a uniform population. Our proof relies on duality techniques, an invariance principle for reversible random walks in a random environment and a convergence result for a system of coalescing random walks in a random environment.
</p>projecteuclid.org/euclid.ejp/1560391565_20190805220401Mon, 05 Aug 2019 22:04 EDTScaling limit of ballistic self-avoiding walk interacting with spatial random permutationshttps://projecteuclid.org/euclid.ejp/1562119474<strong>Volker Betz</strong>, <strong>Lorenzo Taggi</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 37 pp..</p><p><strong>Abstract:</strong><br/>
We consider nearest neighbour spatial random permutations on $\mathbb Z ^{d}$. In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, $\alpha $, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding walk into a background of spatial random permutations, and condition it to cover a macroscopic distance. For large values of $\alpha $ (where long cycles are very unlikely) we show that this walk collapses to a straight line in the scaling limit, and give bounds on the fluctuations that are almost sufficient for diffusive scaling. For proving our results, we develop the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, which may be of independent interest. Among other things, we use them to show exponential decay of correlations for large values of $\alpha $ in great generality.
</p>projecteuclid.org/euclid.ejp/1562119474_20190805220401Mon, 05 Aug 2019 22:04 EDTNonexistence of fractional Brownian fields indexed by cylindershttps://projecteuclid.org/euclid.ejp/1562119475<strong>Nil Venet</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 26 pp..</p><p><strong>Abstract:</strong><br/>
We show in this article that there exists no $H$-fractional Brownian field indexed by the cylinder $\mathbb{S} ^{1} \times ]0,\varepsilon [$ endowed with its product distance $d$ for any $\varepsilon >0$ and $H>0$. This is equivalent to say that $d^{2H}$ is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels – or valid covariances – on the cylinder.
We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also investigate the case of the cylinder endowed with a distance asymptotically close to the product distance in the neighbourhood of a circle.
As a consequence of our result, we show that the set of $H$ such that $d^{2H}$ is negative definite behaves in a discontinuous way with respect to the Gromov-Hausdorff convergence on compact metric spaces.
These results extend our comprehension of kernel construction on metric spaces, and in particular call for alternatives to classical kernels to allow for Gaussian modelling and kernel method learning on cylinders.
</p>projecteuclid.org/euclid.ejp/1562119475_20190805220401Mon, 05 Aug 2019 22:04 EDTProbability tilting of compensated fragmentationshttps://projecteuclid.org/euclid.ejp/1565057003<strong>Quan Shi</strong>, <strong>Alexander R. Watson</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 39 pp..</p><p><strong>Abstract:</strong><br/>
Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. These models are widely used in biology, materials science and nuclear physics, and their asymptotic behaviour at large times is interesting both mathematically and practically. The spine decomposition is a key tool in its study. In this work, we consider the class of compensated fragmentations, or homogeneous growth-fragmentations, recently defined by Bertoin. We give a complete spine decomposition of these processes in terms of a Lévy process with immigration, and apply our result to study the asymptotic properties of the derivative martingale.
</p>projecteuclid.org/euclid.ejp/1565057003_20190805220401Mon, 05 Aug 2019 22:04 EDT