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The Intrinsic geometry of some random manifolds
http://projecteuclid.org/euclid.ecp/1483585770
<strong>Sunder Ram Krishnan</strong>, <strong>Jonathan E. Taylor</strong>, <strong>Robert J. Adler</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.
</p>projecteuclid.org/euclid.ecp/1483585770_20170214040037Tue, 14 Feb 2017 04:00 ESTReal zeros of random Dirichlet serieshttps://projecteuclid.org/euclid.ecp/1568253716<strong>Marco Aymone</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 8 pp..</p><p><strong>Abstract:</strong><br/>
Let $F(\sigma )$ be the random Dirichlet series $F(\sigma )=\sum _{p\in \mathcal{P} } \frac{X_{p}} {p^{\sigma }}$, where $\mathcal{P} $ is an increasing sequence of positive real numbers and $(X_{p})_{p\in \mathcal{P} }$ is a sequence of i.i.d. random variables with $\mathbb{P} (X_{1}=1)=\mathbb{P} (X_{1}=-1)=1/2$. We prove that, for certain conditions on $\mathcal{P} $, if $\sum _{p\in \mathcal{P} }\frac{1} {p}<\infty $ then with positive probability $F(\sigma )$ has no real zeros while if $\sum _{p\in \mathcal{P} }\frac{1} {p}=\infty $, almost surely $F(\sigma )$ has an infinite number of real zeros.
</p>projecteuclid.org/euclid.ecp/1568253716_20190911220221Wed, 11 Sep 2019 22:02 EDTPropagation of chaos for a balls into bins modelhttps://projecteuclid.org/euclid.ecp/1546571102<strong>Nicoletta Cancrini</strong>, <strong>Gustavo Posta</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
Consider a ﬁnite number of balls initially placed in $L$ bins. At each time step a ball is taken from each non-empty bin. Then all the balls are uniformly reassigned into bins. This ﬁnite Markov chain is called Repeated Balls-into-Bins process and is a discrete time interacting particle system with parallel updating. We prove that, starting from a suitable ( chaotic ) set of initial states, as $L\to +\infty $, the numbers of balls in each bin become independent from the rest of the system i.e. we have propagation of chaos . We furthermore study some equilibrium properties of the limiting nonlinear process .
</p>projecteuclid.org/euclid.ecp/1546571102_20190913040515Fri, 13 Sep 2019 04:05 EDTLimit theorems for the tagged particle in exclusion processes on regular treeshttps://projecteuclid.org/euclid.ecp/1548299047<strong>Dayue Chen</strong>, <strong>Peng Chen</strong>, <strong>Nina Gantert</strong>, <strong>Dominik Schmid</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process “seen from the tagged particle” has an ergodic invariant measure.
</p>projecteuclid.org/euclid.ecp/1548299047_20190913040515Fri, 13 Sep 2019 04:05 EDTScaling of the Sasamoto-Spohn model in equilibriumhttps://projecteuclid.org/euclid.ecp/1548817627<strong>Milton Jara</strong>, <strong>Gregorio R. Moreno Flores</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line. The proof, which relies on the second order Boltzmann-Gibbs principle, follows the approach of [9] and does not use any spectral gap argument.
</p>projecteuclid.org/euclid.ecp/1548817627_20190913040515Fri, 13 Sep 2019 04:05 EDTContinuity and growth of free multiplicative convolution semigroupshttps://projecteuclid.org/euclid.ecp/1548817631<strong>Xiaoxue Deng</strong>, <strong>Ping Zhong</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
Let $\mu $ be a compactly supported probability measure on the positive half-line and let $\mu ^{\boxtimes t}$ be the free multiplicative convolution semigroup. We show that the support of $\mu ^{\boxtimes t}$ varies continuously as $t$ changes. We also obtain the asymptotic length of the support of these measures.
</p>projecteuclid.org/euclid.ecp/1548817631_20190913040515Fri, 13 Sep 2019 04:05 EDTHeat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductanceshttps://projecteuclid.org/euclid.ecp/1549357292<strong>Sebastian Andres</strong>, <strong>Jean-Dominique Deuschel</strong>, <strong>Martin Slowik</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 17 pp..</p><p><strong>Abstract:</strong><br/>
We establish heat kernel upper bounds for a continuous-time random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results in [3] to a general class of speed measures. The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure. We also provide a comparison result of this metric with the usual graph distance, which is optimal in the context of the random conductance model with ergodic conductances.
</p>projecteuclid.org/euclid.ecp/1549357292_20190913040515Fri, 13 Sep 2019 04:05 EDTErratum: Nonconventional random matrix productshttps://projecteuclid.org/euclid.ecp/1549530018<strong>Yuri Kifer</strong>, <strong>Sasha Sodin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 1 pp..</p><p><strong>Abstract:</strong><br/>
The proof of Theorem 2.3 in our paper [3] is fully justified only under the additional assumption $q_i(n)=a_in+b_i,\, i=1,...,\ell $.
</p>projecteuclid.org/euclid.ecp/1549530018_20190913040515Fri, 13 Sep 2019 04:05 EDTExponential convergence to equilibrium for the $d$-dimensional East modelhttps://projecteuclid.org/euclid.ecp/1568361881<strong>Laure Marêché</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
Kinetically constrained models (KCMs) are interacting particle systems on $\mathbb{Z} ^{d}$ with a continuous-time constrained Glauber dynamics, which were introduced by physicists to model the liquid-glass transition. One of the most well-known KCMs is the one-dimensional East model. Its generalization to higher dimension, the $d$-dimensional East model, is much less understood. Prior to this paper, convergence to equilibrium in the $d$-dimensional East model was proven to be at least stretched exponential, by Chleboun, Faggionato and Martinelli in 2015. We show that the $d$-dimensional East model exhibits exponential convergence to equilibrium in all settings for which convergence is possible.
</p>projecteuclid.org/euclid.ecp/1568361881_20190913040515Fri, 13 Sep 2019 04:05 EDTBounds for distances and geodesic dimension in Liouville first passage percolationhttps://projecteuclid.org/euclid.ecp/1568361882<strong>Ewain Gwynne</strong>, <strong>Joshua Pfeffer</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
For $\xi \geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\varepsilon \mathbb{Z} ^{2}$ obtained by weighting each vertex by $\varepsilon e^{\xi h_{\varepsilon }(z)}$, where $h_{\varepsilon }(z)$ is the average of the whole-plane Gaussian free field $h$ over the circle $\partial B_{\varepsilon }(z)$. Ding and Gwynne (2018) showed that for $\gamma \in (0,2)$, LFPP with parameter $\xi = \gamma /d_{\gamma }$ is related to $\gamma $-Liouville quantum gravity (LQG), where $d_{\gamma }$ is the $\gamma $-LQG dimension exponent. For $\xi > 2/d_{2}$, LFPP is instead expected to be related to LQG with central charge greater than 1. We prove several estimates for LFPP distances for general $\xi \geq 0$. For $\xi \leq 2/d_{2}$, this leads to new bounds for $d_{\gamma }$ which improve on the best previously known upper (resp. lower) bounds for $d_{\gamma }$ in the case when $\gamma > \sqrt{8/3} $ (resp. $\gamma \in (0.4981, \sqrt{8/3} )$). These bounds are consistent with the Watabiki (1993) prediction for $d_{\gamma }$. However, for $\xi > 1/\sqrt{3} $ (or equivalently for LQG with central charge larger than 17) our bounds are inconsistent with the analytic continuation of Watabiki’s prediction to the $\xi >2/d_{2}$ regime. We also obtain an upper bound for the Euclidean dimension of LFPP geodesics.
</p>projecteuclid.org/euclid.ecp/1568361882_20190913040515Fri, 13 Sep 2019 04:05 EDTOn the eigenvalues of truncations of random unitary matriceshttps://projecteuclid.org/euclid.ecp/1568361883<strong>Elizabeth Meckes</strong>, <strong>Kathryn Stewart</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Réffy identified the limiting spectral measure if $\frac{m} {n}\to \alpha $, as $n\to \infty $; under suitable scaling, the family $\{\mu _{\alpha }\}_{\alpha \in (0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $\alpha $) and uniform measure on the unit circle (as $\alpha \to 1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $\mu _{\alpha }$ is typically of order $\sqrt{\frac {\log (m)}{m}} $ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new “Coulomb transport inequality” due to Chafaï, Hardy, and Maïda.
</p>projecteuclid.org/euclid.ecp/1568361883_20190913040515Fri, 13 Sep 2019 04:05 EDTOn the tails of the limiting QuickSort densityhttps://projecteuclid.org/euclid.ecp/1550113298<strong>James Allen Fill</strong>, <strong>Wei-Chun Hung</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We give upper and lower asymptotic bounds for the left tail and for the right tail of the continuous limiting QuickSort density $f$ that are nearly matching in each tail. The bounds strengthen results from a paper of Svante Janson (2015) concerning the corresponding distribution function $F$. Furthermore, we obtain similar bounds on absolute values of derivatives of $f$ of each order.
</p>projecteuclid.org/euclid.ecp/1550113298_20190918040052Wed, 18 Sep 2019 04:00 EDTDean-Kawasaki dynamics: ill-posedness vs. trivialityhttps://projecteuclid.org/euclid.ecp/1550113299<strong>Vitalii Konarovskyi</strong>, <strong>Tobias Lehmann</strong>, <strong>Max-K. von Renesse</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
We prove that the Dean-Kawasaki SPDE admits a solution only in integer parameter regimes, in which case the solution is given in terms of a system of non-interacting particles.
</p>projecteuclid.org/euclid.ecp/1550113299_20190918040052Wed, 18 Sep 2019 04:00 EDTAlmost sure limit theorems on Wiener chaos: the non-central casehttps://projecteuclid.org/euclid.ecp/1550199821<strong>Ehsan Azmoodeh</strong>, <strong>Ivan Nourdin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Itô integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].
</p>projecteuclid.org/euclid.ecp/1550199821_20190918040052Wed, 18 Sep 2019 04:00 EDTCritical percolation and the incipient infinite cluster on Galton-Watson treeshttps://projecteuclid.org/euclid.ecp/1550480494<strong>Marcus Michelen</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 13 pp..</p><p><strong>Abstract:</strong><br/>
We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Galton-Watson tree and we prove a limit law for the number of vertices in the IIC at depth $n$, again depending only on the offspring distribution. Provided the offspring distribution used to generate these Galton-Watson trees has all finite moments, each of these results holds almost-surely.
</p>projecteuclid.org/euclid.ecp/1550480494_20190918040052Wed, 18 Sep 2019 04:00 EDTExpectation of the largest bet size in the Labouchere systemhttps://projecteuclid.org/euclid.ecp/1550826036<strong>Yanjun Han</strong>, <strong>Guanyang Wang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
For the Labouchere system with winning probability $p$ at each coup, we prove that the expectation of the largest bet size under any initial list is finite if $p>\frac{1} {2}$, and is infinite if $p\le \frac{1} {2}$, solving the open conjecture in [6]. The same result holds for a general family of betting systems, and the proof builds upon a recursive representation of the optimal betting system in the larger family.
</p>projecteuclid.org/euclid.ecp/1550826036_20190918040052Wed, 18 Sep 2019 04:00 EDTConcentration for Coulomb gases on compact manifoldshttps://projecteuclid.org/euclid.ecp/1553133701<strong>David García-Zelada</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 18 pp..</p><p><strong>Abstract:</strong><br/>
We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafaï, Hardy and Maïda on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.
</p>projecteuclid.org/euclid.ecp/1553133701_20190918040052Wed, 18 Sep 2019 04:00 EDTClosed-form formulas for the distribution of the jumps of doubly-stochastic Poisson processeshttps://projecteuclid.org/euclid.ecp/1553133702<strong>Arturo Valdivia</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the obtainment of closed-form formulas for the distribution of the jumps of a doubly-stochastic Poisson process. The problem is approached in two ways. On the one hand, we translate the problem to the computation of multiple derivatives of the Hazard process cumulant generating function; this leads to a closed-form formula written in terms of Bell polynomials. On the other hand, for Hazard processes driven by Lévy processes, we use Malliavin calculus in order to express the aforementioned distributions in an appealing recursive manner. We outline the potential application of these results in credit risk.
</p>projecteuclid.org/euclid.ecp/1553133702_20190918040052Wed, 18 Sep 2019 04:00 EDTA Hoeffding inequality for Markov chainshttps://projecteuclid.org/euclid.ecp/1553133703<strong>Shravas Rao</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We prove deviation bounds for the random variable $\sum _{i=1}^{n} f_i(Y_i)$ in which $\{Y_i\}_{i=1}^{\infty }$ is a Markov chain with stationary distribution and state space $[N]$, and $f_i: [N] \rightarrow [-a_i, a_i]$. Our bound improves upon previously known bounds in that the dependence is on $\sqrt{a_1^2+\cdots +a_n^2} $ rather than $\max _{i}\{a_i\}\sqrt{n} .$ We also prove deviation bounds for certain types of sums of vector–valued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten $\infty $-norm of a random matrix whose entries are obtained from a Markov chain.
</p>projecteuclid.org/euclid.ecp/1553133703_20190918040052Wed, 18 Sep 2019 04:00 EDTA spectral decomposition for a simple mutation modelhttps://projecteuclid.org/euclid.ecp/1553133704<strong>Martin Möhle</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 14 pp..</p><p><strong>Abstract:</strong><br/>
We consider a population of $N$ individuals. Each individual has a type belonging to some at most countable type space $K$. At each time step each individual of type $k\in K$ mutates to type $l\in K$ independently of the other individuals with probability $m_{k,l}$. It is shown that the associated empirical measure process is Markovian. For the two-type case $K=\{0,1\}$ we derive an explicit spectral decomposition for the transition matrix $P$ of the Markov chain $Y=(Y_n)_{n\ge 0}$, where $Y_n$ denotes the number of individuals of type $1$ at time $n$. The result in particular shows that $P$ has eigenvalues $(1-m_{0,1}-m_{1,0})^i$, $i\in \{0,\ldots ,N\}$. Applications to mean first passage times are provided.
</p>projecteuclid.org/euclid.ecp/1553133704_20190918040052Wed, 18 Sep 2019 04:00 EDTDiscrete harmonic functions in Lipschitz domainshttps://projecteuclid.org/euclid.ecp/1568793625<strong>Sami Mustapha</strong>, <strong>Mohamed Sifi</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 15 pp..</p><p><strong>Abstract:</strong><br/>
We prove the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in $\mathbb{Z} ^{d}$. Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.
</p>projecteuclid.org/euclid.ecp/1568793625_20190918040052Wed, 18 Sep 2019 04:00 EDTImproved order 1/4 convergence for piecewise constant policy approximation of stochastic control problemshttps://projecteuclid.org/euclid.ecp/1568793626<strong>Espen R. Jakobsen</strong>, <strong>Athena Picarelli</strong>, <strong>Christoph Reisinger</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
In N. V. Krylov, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electron. J. Probab., 4(2), 1999 , it is proved under standard assumptions that the value functions of controlled diffusion processes can be approximated with order 1/6 error by those with controls which are constant on uniform time intervals. In this note we refine the proof and show that the provable rate can be improved to 1/4, which is optimal in our setting. Moreover, we demonstrate the improvements this implies for error estimates derived by similar techniques for approximation schemes, bringing these in line with the best available results from the PDE literature.
</p>projecteuclid.org/euclid.ecp/1568793626_20190918040052Wed, 18 Sep 2019 04:00 EDTError bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector modelshttps://projecteuclid.org/euclid.ecp/1553220032<strong>Lê Vǎn Thành</strong>, <strong>Nguyen Ngoc Tu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
This paper gives the Kolmogorov and Wasserstein bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models. The Kolmogorov bound is new while the Wasserstein bound improves a result obtained recently by Kirkpatrick and Nawaz [Journal of Statistical Physics, 165 (2016), no. 6, 1114–1140]. The proof is based on Stein’s method for exchangeable pairs.
</p>projecteuclid.org/euclid.ecp/1553220032_20190930220914Mon, 30 Sep 2019 22:09 EDTConditions for the finiteness of the moments of the volume of level setshttps://projecteuclid.org/euclid.ecp/1553220033<strong>D. Armentano</strong>, <strong>J-M. Azaïs</strong>, <strong>D. Ginsbourger</strong>, <strong>J.R. León</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 8 pp..</p><p><strong>Abstract:</strong><br/>
Let $X(t)$ be a Gaussian random field $\mathbb R^d\to \mathbb R$. Using the notion of $(d-1)$- integral geometric measures , we establish a relation between (a) the volume of level sets, and (b) the number of crossings of the restriction of the random field to a line. Using this relation we prove the equivalence between the finiteness of the expectation and the finiteness of the second spectral moment matrix. Sufficient conditions for finiteness of higher moments are also established.
</p>projecteuclid.org/euclid.ecp/1553220033_20190930220914Mon, 30 Sep 2019 22:09 EDTCritical Liouville measure as a limit of subcritical measureshttps://projecteuclid.org/euclid.ecp/1553306557<strong>Juhan Aru</strong>, <strong>Ellen Powell</strong>, <strong>Avelio Sepúlveda</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 16 pp..</p><p><strong>Abstract:</strong><br/>
We study how the Gaussian multiplicative chaos (GMC) measures $\mu ^\gamma $ corresponding to the 2D Gaussian free field change when $\gamma $ approaches the critical parameter $2$. In particular, we show that as $\gamma \to 2^{-}$, $(2-\gamma )^{-1}\mu ^\gamma $ converges in probability to $2\mu '$, where $\mu '$ is the critical GMC measure.
</p>projecteuclid.org/euclid.ecp/1553306557_20190930220914Mon, 30 Sep 2019 22:09 EDTSubsequential tightness of the maximum of two dimensional Ginzburg-Landau fieldshttps://projecteuclid.org/euclid.ecp/1553306561<strong>Wei Wu</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We prove the subsequential tightness of centered maxima of two-dimensional Ginzburg-Landau fields with bounded elliptic contrast.
</p>projecteuclid.org/euclid.ecp/1553306561_20190930220914Mon, 30 Sep 2019 22:09 EDTOn coupling and “vacant set level set” percolationhttps://projecteuclid.org/euclid.ecp/1554170646<strong>Alain-Sol Sznitman</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this note we discuss “vacant set level set” percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpins a certain strict monotonicity property that has significant consequences for the percolation diagram. It is presently open whether a similar looking diagram holds in the case of ${\mathbb Z}^d$, $d \ge 3$.
</p>projecteuclid.org/euclid.ecp/1554170646_20190930220914Mon, 30 Sep 2019 22:09 EDTA note on transportation cost inequalities for diffusions with reflectionshttps://projecteuclid.org/euclid.ecp/1554429763<strong>Soumik Pal</strong>, <strong>Andrey Sarantsev</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusion processes with suitable drift and diffusion coefficients. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.
</p>projecteuclid.org/euclid.ecp/1554429763_20190930220914Mon, 30 Sep 2019 22:09 EDTSimultaneous boundary hitting by coupled reflected Brownian motionshttps://projecteuclid.org/euclid.ecp/1555034600<strong>Krzysztof Burdzy</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.
</p>projecteuclid.org/euclid.ecp/1555034600_20190930220914Mon, 30 Sep 2019 22:09 EDTRandom replacements in Pólya urns with infinitely many colourshttps://projecteuclid.org/euclid.ecp/1556589626<strong>Svante Janson</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We consider the general version of Pólya urns recently studied by Bandyopadhyay and Thacker (2016+) and Mailler and Marckert (2017), with the space of colours being any Borel space $S$ and the state of the urn being a finite measure on $S$. We consider urns with random replacements, and show that these can be regarded as urns with deterministic replacements using the colour space $S\times [0,1]$.
</p>projecteuclid.org/euclid.ecp/1556589626_20190930220914Mon, 30 Sep 2019 22:09 EDTExit boundaries of multidimensional SDEshttps://projecteuclid.org/euclid.ecp/1558339220<strong>Russell Lyons</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 2 pp..</p><p><strong>Abstract:</strong><br/>
We show that solutions to multidimensional SDEs with Lipschitz coefficients and driven by Brownian motion never reach the set where all coefficients vanish unless the initial position belongs to that set.
</p>projecteuclid.org/euclid.ecp/1558339220_20190930220914Mon, 30 Sep 2019 22:09 EDTAn infinite-dimensional helix invariant under spherical projectionshttps://projecteuclid.org/euclid.ecp/1559354658<strong>Zakhar Kabluchko</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 13 pp..</p><p><strong>Abstract:</strong><br/>
We classify all subsets $S$ of the projective Hilbert space with the following property: for every point $\pm s_{0}\in S$, the spherical projection of $S\backslash \{\pm s_{0}\}$ on the hyperplane orthogonal to $\pm s_{0}$ is isometric to $S\backslash \{\pm s_{0}\}$. In probabilistic terms, this means that we characterize all zero-mean Gaussian processes $Z=(Z(t))_{t\in T}$ with the property that for every $s_{0}\in T$ the conditional distribution of $(Z(t))_{t\in T}$ given that $Z(s_{0})=0$ coincides with the distribution of $(\varphi (t; s_{0}) Z(t))_{t\in T}$ for some function $\varphi (t;s_{0})$. A basic example of such process is the stationary zero-mean Gaussian process $(X(t))_{t\in \mathbb{R} }$ with covariance function $\mathbb E [X(s) X(t)] = 1/\cosh (t-s)$. We show that, in general, the process $Z$ can be decomposed into a union of mutually independent processes of two types: (i) processes of the form $(a(t) X(\psi (t)))_{t\in T}$, with $a: T\to \mathbb{R} $, $\psi (t): T\to \mathbb{R} $, and (ii) certain exceptional Gaussian processes defined on four-point index sets. The above problem is reduced to the classification of metric spaces in which in every triangle the largest side equals the sum of the remaining two sides.
</p>projecteuclid.org/euclid.ecp/1559354658_20190930220914Mon, 30 Sep 2019 22:09 EDTThe frog model on trees with drifthttps://projecteuclid.org/euclid.ecp/1559354659<strong>Erin Beckman</strong>, <strong>Natalie Frank</strong>, <strong>Yufeng Jiang</strong>, <strong>Matthew Junge</strong>, <strong>Si Tang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a $d$-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. Additionally, we describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one. This implies that the critical drift has a limit as $d$ tends to infinity along certain subsequences.
</p>projecteuclid.org/euclid.ecp/1559354659_20190930220914Mon, 30 Sep 2019 22:09 EDTSome conditional limiting theorems for symmetric Markov processes with tightness propertyhttps://projecteuclid.org/euclid.ecp/1569895735<strong>Guoman He</strong>, <strong>Ge Yang</strong>, <strong>Yixia Zhu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
Let $X$ be an $\mu $-symmetric irreducible Markov process on $I$ with strong Feller property. In addition, we assume that $X$ possesses a tightness property. In this paper, we prove some conditional limiting theorems for the process $X$. The emphasis is on conditional ergodic theorem. These results are also discussed in the framework of one-dimensional diffusions.
</p>projecteuclid.org/euclid.ecp/1569895735_20190930220914Mon, 30 Sep 2019 22:09 EDTBi-log-concavity: some properties and some remarks towards a multi-dimensional extensionhttps://projecteuclid.org/euclid.ecp/1569895736<strong>Adrien Saumard</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 8 pp..</p><p><strong>Abstract:</strong><br/>
Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measure with a log-concave one is bi-log-concave. Consequently, infinitely differentiable, positive densities are dense in the set of bi-log-concave densities for $ L_{p}$-norms, $p\in \left [1,+\infty \right ]$. We also derive a necessary and sufficient condition for the convolution of two bi-log-concave measures to be bi-log-concave. We conclude this note by discussing a way of defining a multi-dimensional extension of the notion of bi-log-concavity.
</p>projecteuclid.org/euclid.ecp/1569895736_20190930220914Mon, 30 Sep 2019 22:09 EDTThe bullet problem with discrete speedshttps://projecteuclid.org/euclid.ecp/1559700463<strong>Brittany Dygert</strong>, <strong>Christoph Kinzel</strong>, <strong>Matthew Junge</strong>, <strong>Annie Raymond</strong>, <strong>Erik Slivken</strong>, <strong>Jennifer Zhu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
Bullets are fired from the origin of the positive real line, one per second, with independent speeds sampled uniformly from a discrete set. Collisions result in mutual annihilation. We show that a bullet with the second largest speed survives with positive probability, while a bullet with the smallest speed does not. This also holds for exponential spacings between firing times. Our results imply that the middle-velocity particle survives with positive probability in a two-sided version of the bullet process with three speeds known to physicists as ballistic annihilation.
</p>projecteuclid.org/euclid.ecp/1559700463_20191007220528Mon, 07 Oct 2019 22:05 EDTImproved Hölder continuity near the boundary of one-dimensional super-Brownian motionhttps://projecteuclid.org/euclid.ecp/1559700464<strong>Jieliang Hong</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We show that the local time of one-dimensional super-Brownian motion is locally $\gamma $-Hölder continuous near the boundary if $0<\gamma <3$ and fails to be locally $\gamma $-Hölder continuous if $\gamma >3$.
</p>projecteuclid.org/euclid.ecp/1559700464_20191007220528Mon, 07 Oct 2019 22:05 EDTSensitivity of the frog model to initial conditionshttps://projecteuclid.org/euclid.ecp/1559700465<strong>Tobias Johnson</strong>, <strong>Leonardo T. Rolla</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
The frog model is an interacting particle system on a graph. Active particles perform independent simple random walks, while sleeping particles remain inert until visited by an active particle. Some number of sleeping particles are placed at each site sampled independently from a certain distribution, and then one particle is activated to begin the process. We show that the recurrence or transience of the model is sensitive not just to the expectation but to the entire distribution. This is in contrast to closely related models like branching random walk and activated random walk.
</p>projecteuclid.org/euclid.ecp/1559700465_20191007220528Mon, 07 Oct 2019 22:05 EDTRigidity for zero sets of Gaussian entire functionshttps://projecteuclid.org/euclid.ecp/1559700466<strong>Avner Kiro</strong>, <strong>Alon Nishry</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane.
We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kernel, the zero set is “fully rigid”. This means that if we know the location of the zeros in the complement of any given compact set, then the number and location of the zeros inside that set can be determined uniquely. As far as we are aware, this is the first explicit construction in a natural class of random point processes with full rigidity.
</p>projecteuclid.org/euclid.ecp/1559700466_20191007220528Mon, 07 Oct 2019 22:05 EDTGrowing in time IDLA cluster is recurrenthttps://projecteuclid.org/euclid.ecp/1560391477<strong>Ruojun Huang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We show that Internal Diffusion Limited Aggregation (IDLA )on $\mathbb{Z} ^{d}$ has near optimal Cheeger constant when the growing cluster is large enough. This implies, through a heat kernel lower bound derived previously in [11], that simple random walk evolving independently on growing in time IDLA cluster is recurrent when $d\ge 3$.
</p>projecteuclid.org/euclid.ecp/1560391477_20191007220528Mon, 07 Oct 2019 22:05 EDTAn upper bound for the probability of visiting a distant point by a critical branching random walk in $\mathbb{Z} ^{4}$https://projecteuclid.org/euclid.ecp/1560477644<strong>Qingsan Zhu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 6 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we study the probability of visiting a distant point $a\in \mathbb{Z} ^{4}$ by a critical branching random walk starting at the origin. We prove that this probability is bounded by $1/(|a|^{2}\log |a|)$ up to a constant factor.
</p>projecteuclid.org/euclid.ecp/1560477644_20191007220528Mon, 07 Oct 2019 22:05 EDTOn the martingale property in the rough Bergomi modelhttps://projecteuclid.org/euclid.ecp/1560477645<strong>Paul Gassiat</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation $\rho $ between the driving Brownian motions of the stock and the volatility is nonpositive. We also show that for each $\rho <0$ and $m> \frac{1} {{1-\rho ^{2}}}$, the $m$-th moment of the stock price is infinite at each positive time.
</p>projecteuclid.org/euclid.ecp/1560477645_20191007220528Mon, 07 Oct 2019 22:05 EDTBerry-Esseen bounds in the Breuer-Major CLT and Gebelein’s inequalityhttps://projecteuclid.org/euclid.ecp/1561169050<strong>Ivan Nourdin</strong>, <strong>Giovanni Peccati</strong>, <strong>Xiaochuan Yang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We derive explicit Berry-Esseen bounds in the total variation distance for the Breuer-Major central limit theorem, in the case of a subordinating function $\varphi $ satisfying minimal regularity assumptions. Our approach is based on the combination of the Malliavin-Stein approach for normal approximations with Gebelein’s inequality, bounding the covariance of functionals of Gaussian fields in terms of maximal correlation coefficients.
</p>projecteuclid.org/euclid.ecp/1561169050_20191007220528Mon, 07 Oct 2019 22:05 EDTLarge deviations of the long term distribution of a non Markov processhttps://projecteuclid.org/euclid.ecp/1561169051<strong>Anatolii Puhalskii</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We prove that the long term distribution of the queue length process in an ergodic generalised Jackson network obeys the Large Deviation Principle with a deviation function given by the quasipotential. The latter is related to the unique long term idempotent distribution, which is also a stationary idempotent distribution, of the large deviation limit of the queue length process. The proof draws on developments in queueing network stability and idempotent probability.
</p>projecteuclid.org/euclid.ecp/1561169051_20191007220528Mon, 07 Oct 2019 22:05 EDTKemeny’s constant for one-dimensional diffusionshttps://projecteuclid.org/euclid.ecp/1561169052<strong>Ross Pinsky</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 5 pp..</p><p><strong>Abstract:</strong><br/>
Let $X(\cdot )$ be a non-degenerate, positive recurrent one-dimensional diffusion process on $\mathbb{R} $ with invariant probability density $\mu (x)$, and let $\tau _{y}=\inf \{t\ge 0: X(t)=y\}$ denote the first hitting time of $y$. Let $\mathcal{X} $ be a random variable independent of the diffusion process $X(\cdot )$ and distributed according to the process’s invariant probability measure $\mu (x)dx$. Denote by $\mathcal{E} ^{\mu }$ the expectation with respect to $\mathcal{X} $. Consider the expression \[ \mathcal{E} ^{\mu }E_{x}\tau _{\mathcal{X} }=\int _{-\infty }^{\infty }(E_{x}\tau _{y})\mu (y)dy, \ x\in \mathbb{R} . \] In words, this expression is the expected hitting time of the diffusion starting from $x$ of a point chosen randomly according to the diffusion’s invariant distribution. We show that this expression is constant in $x$, and that it is finite if and only if $\pm \infty $ are entrance boundaries for the diffusion. This result generalizes to diffusion processes the corresponding result in the setting of finite Markov chains, where the constant value is known as Kemeny’s constant.
</p>projecteuclid.org/euclid.ecp/1561169052_20191007220528Mon, 07 Oct 2019 22:05 EDTRotatable random sequences in local fieldshttps://projecteuclid.org/euclid.ecp/1561169053<strong>Steven N. Evans</strong>, <strong>Daniel Raban</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
An infinite sequence of real random variables $(\xi _{1}, \xi _{2}, \dots )$ is said to be rotatable if every finite subsequence $(\xi _{1}, \dots , \xi _{n})$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(\xi _{1}, \xi _{2}, \dots )$ is rotatable if and only if $\xi _{j} = \tau \eta _{j}$ for all $j$, where $(\eta _{1}, \eta _{2}, \dots )$ is a sequence of independent standard Gaussian random variables and $\tau $ is an independent nonnegative random variable. Freedman’s theorem is equivalent to a classical result of Schoenberg which says that a continuous function $\phi : \mathbb{R} _{+} \to \mathbb{C} $ with $\phi (0) = 1$ is completely monotone if and only if $\phi _{n}: \mathbb{R} ^{n} \to \mathbb{R} $ given by $\phi _{n}(x_{1}, \ldots , x_{n}) = \phi (x_{1}^{2} + \cdots + x_{n}^{2})$ is nonnegative definite for all $n \in \mathbb{N} $. We establish the analogue of Freedman’s theorem for sequences of random variables taking values in local fields using probabilistic methods and then use it to establish a local field analogue of Schoenberg’s result. Along the way, we obtain a local field counterpart of an observation variously attributed to Maxwell, Poincaré, and Borel which says that if $(\zeta _{1}, \ldots , \zeta _{n})$ is uniformly distributed on the sphere of radius $\sqrt{n} $ in $\mathbb{R} ^{n}$, then, for fixed $k \in \mathbb{N} $, the distribution of $(\zeta _{1}, \ldots , \zeta _{k})$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n \to \infty $.
</p>projecteuclid.org/euclid.ecp/1561169053_20191007220528Mon, 07 Oct 2019 22:05 EDTWeighted graphs and complex Gaussian free fieldshttps://projecteuclid.org/euclid.ecp/1561169054<strong>Gregory F. Lawler</strong>, <strong>Petr Panov</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
We prove a combinatorial statement about the distribution of directed currents in a complex “loop soup” and use it to give a new proof of the isomorphism, which relates loop measures and complex Gaussian free fields.
</p>projecteuclid.org/euclid.ecp/1561169054_20191007220528Mon, 07 Oct 2019 22:05 EDTPhase transitions for edge-reinforced random walks on the half-linehttps://projecteuclid.org/euclid.ecp/1561169055<strong>Jiro Akahori</strong>, <strong>Andrea Collevecchio</strong>, <strong>Masato Takei</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the behaviour of a class of edge-reinforced random walks on $\mathbb{Z} _{+}$, with heterogeneous initial weights, where each edge weight can be updated only when the edge is traversed from left to right. We provide a description for different behaviours of this process and describe phase transitions that arise as trade-offs between the strength of the reinforcement and that of the initial weights. Our result aims to complete the ones given by Davis [3, 4], Takeshima [9, 10] and Vervoort [11].
</p>projecteuclid.org/euclid.ecp/1561169055_20191007220528Mon, 07 Oct 2019 22:05 EDTMartingale spaces and representations under absolutely continuous changes of probabilityhttps://projecteuclid.org/euclid.ecp/1570500302<strong>Anna Aksamit</strong>, <strong>Claudio Fontana</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 13 pp..</p><p><strong>Abstract:</strong><br/>
In a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale representation property (MRP) is always stable under locally absolutely continuous changes of probability. Our approach relies on minimal requirements, is constructive and, as shown by a simple example, enables us to study situations which cannot be covered by the existing theory.
</p>projecteuclid.org/euclid.ecp/1570500302_20191007220528Mon, 07 Oct 2019 22:05 EDTOn Markovian random networkshttps://projecteuclid.org/euclid.ecp/1562119368<strong>Yves Le Jan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 7 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the relation to random configurations and combinatorial maps of the Eulerian networks defined by Poissonian emsembles of Markov loops.
</p>projecteuclid.org/euclid.ecp/1562119368_20191011220121Fri, 11 Oct 2019 22:01 EDTConvergence of complex martingale for a branching random walk in a time random environmenthttps://projecteuclid.org/euclid.ecp/1562119369<strong>Xiaoqiang Wang</strong>, <strong>Chunmao Huang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 14 pp..</p><p><strong>Abstract:</strong><br/>
We consider a discrete-time branching random walk in a stationary and ergodic environment $\xi =(\xi _{n})$ indexed by time $n\in \mathbb{N} $. Let $W_{n}(z)$ ($z\in \mathbb{C} ^{d}$) be the natural complex martingale of the process. We show sufficient conditions for its almost sure and quenched $L^{\alpha }$ convergence, as well as the existence of quenched moments and weighted moments of its limit, and also describe the exponential convergence rate.
</p>projecteuclid.org/euclid.ecp/1562119369_20191011220121Fri, 11 Oct 2019 22:01 EDTAbsolute continuity of the martingale limit in branching processes in random environmenthttps://projecteuclid.org/euclid.ecp/1562119370<strong>Ewa Damek</strong>, <strong>Nina Gantert</strong>, <strong>Konrad Kolesko</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 13 pp..</p><p><strong>Abstract:</strong><br/>
We consider a supercritical branching process $Z_{n}$ in a stationary and ergodic random environment $\xi =(\xi _{n})_{n\ge 0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_{n}=Z_{n}/(\mathbb{E} [Z_{n}|\xi ])$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $\xi $ the law of $W$ conditioned on the environment $\xi $ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of [10], and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of $W$.
</p>projecteuclid.org/euclid.ecp/1562119370_20191011220121Fri, 11 Oct 2019 22:01 EDTProjections of scaled Bessel processshttps://projecteuclid.org/euclid.ecp/1562119371<strong>Constantinos Kardaras</strong>, <strong>Johannes Ruf</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the process $s (X+Y)$ is a local martingale. We study the representation and the dynamics of $s(X+Y)$, projected on the filtration generated by $X$. This projection is a strict supermartingale if, and only if, $m<2$. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of $X$ at zero.
</p>projecteuclid.org/euclid.ecp/1562119371_20191011220121Fri, 11 Oct 2019 22:01 EDTProbability to be positive for the membrane model in dimensions 2 and 3https://projecteuclid.org/euclid.ecp/1562292105<strong>Simon Buchholz</strong>, <strong>Jean-Dominique Deuschel</strong>, <strong>Noemi Kurt</strong>, <strong>Florian Schweiger</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 14 pp..</p><p><strong>Abstract:</strong><br/>
We consider the membrane model on a box $V_{N}\subset \mathbb{Z} ^{n}$ of size $(2N+1)^{n}$ with zero boundary condition in the subcritical dimensions $n=2$ and $n=3$. We show optimal estimates for the probability that the field is positive in a subset $D_{N}$ of $V_{N}$. In particular we obtain for $D_{N}=V_{N}$ that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area $N^{n-1}$.
</p>projecteuclid.org/euclid.ecp/1562292105_20191011220121Fri, 11 Oct 2019 22:01 EDTUpper tail large deviations in Brownian directed percolationhttps://projecteuclid.org/euclid.ecp/1562292106<strong>Christopher Janjigian</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof of a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.
</p>projecteuclid.org/euclid.ecp/1562292106_20191011220121Fri, 11 Oct 2019 22:01 EDTOn the existence of continuous processes with given one-dimensional distributionshttps://projecteuclid.org/euclid.ecp/1566957630<strong>Luca Pratelli</strong>, <strong>Pietro Rigo</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{P} $ be the collection of Borel probability measures on $\mathbb{R} $, equipped with the weak topology, and let $\mu :[0,1]\rightarrow \mathcal{P} $ be a continuous map. Say that $\mu $ is presentable if $X_{t}\sim \mu _{t}$, $t\in [0,1]$, for some real process $X$ with continuous paths. It may be that $\mu $ fails to be presentable. Hence, firstly, conditions for presentability are given. For instance, $\mu $ is presentable if $\mu _{t}$ is supported by an interval (possibly, by a singleton) for all but countably many $t$. Secondly, assuming $\mu $ presentable, we investigate whether the quantile process $Q$ induced by $\mu $ has continuous paths. The latter is defined, on the probability space $((0,1),\mathcal{B} (0,1),\mbox{Lebesgue measure} )$, by \[ Q_{t}(\alpha )=\inf \, \bigl \{x\in \mathbb{R} :\mu _{t}(-\infty ,x]\ge \alpha \bigl \} \quad \quad \mbox{for all } t\in [0,1]\mbox{ and } \alpha \in (0,1). \] A few open problems are discussed as well.
</p>projecteuclid.org/euclid.ecp/1566957630_20191011220121Fri, 11 Oct 2019 22:01 EDTRandom walk in a stratified independent random environmenthttps://projecteuclid.org/euclid.ecp/1567649074<strong>Brémont Julien</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 15 pp..</p><p><strong>Abstract:</strong><br/>
We study Markov chains on a lattice in a codimension-one stratified independent random environment, exploiting results previously established in [2]. The random walk is first shown to be transient in dimension at least three. Focusing on dimension two, we provide sharp sufficient conditions for either recurrence or transience. We determine the critical scale of the local drift along the strata, corresponding to the frontier between the two regimes.
</p>projecteuclid.org/euclid.ecp/1567649074_20191011220121Fri, 11 Oct 2019 22:01 EDTVariational estimates for martingale paraproductshttps://projecteuclid.org/euclid.ecp/1568253710<strong>Vjekoslav Kovač</strong>, <strong>Pavel Zorin-Kranich</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 14 pp..</p><p><strong>Abstract:</strong><br/>
We show that bilinear variational estimates of Do, Muscalu, and Thiele [7] remain valid for a pair of general martingales with respect to the same filtration. Our result can also be viewed as an off-diagonal generalization of the Burkholder–Davis–Gundy inequality for martingale rough paths by Chevyrev and Friz [4].
</p>projecteuclid.org/euclid.ecp/1568253710_20191011220121Fri, 11 Oct 2019 22:01 EDTExistence and uniqueness of solution to scalar BSDEs with $L\exp (\mu \sqrt{2\log (1+L)} )$-integrable terminal values: the critical casehttps://projecteuclid.org/euclid.ecp/1568253711<strong>Shengjun Fan</strong>, <strong>Ying Hu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
In [8], the existence of the solution is proved for a scalar linearly growingbackward stochastic differential equation (BSDE) when the terminal value is$L\exp (\mu \sqrt{2\log (1+L)} )$-integrable for a positive parameter $\mu >\mu _{0}$ with a critical value $\mu _{0}$, and a counterexample is provided to show that the preceding integrability for $\mu <\mu _{0}$ is not sufficient to guarantee the existence of the solution. Afterwards, the uniqueness result (with $\mu >\mu _{0}$) is also given in [3] for the preceding BSDE under the uniformly Lipschitz condition of the generator. In this note, we prove that these two results still hold for the critical case: $\mu =\mu _{0}$.
</p>projecteuclid.org/euclid.ecp/1568253711_20191011220121Fri, 11 Oct 2019 22:01 EDTOptimal stopping of oscillating Brownian motionhttps://projecteuclid.org/euclid.ecp/1568253712<strong>Ernesto Mordecki</strong>, <strong>Paavo Salminen</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We solve optimal stopping problems for an oscillating Brownian motion, i.e. a diffusion with positive piecewise constant volatility changing at the point $x=0$. Let $\sigma _{1}$ and $\sigma _{2}$ denote the volatilities on the negative and positive half-lines, respectively. Our main result is that continuation region of the optimal stopping problem with reward $((1+x)^{+})^{2}$ can be disconnected for some values of the discount rate when $2\sigma _{1}^{2}<\sigma _{2}^{2}$. Based on the fact that the skew Brownian motion in natural scale is an oscillating Brownian motion, the obtained results are translated into corresponding results for the skew Brownian motion.
</p>projecteuclid.org/euclid.ecp/1568253712_20191011220121Fri, 11 Oct 2019 22:01 EDTConvergence of point processes associated with coupon collector’s and Dixie cup problemshttps://projecteuclid.org/euclid.ecp/1568253713<strong>Andrii Ilienko</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
We prove that, in the coupon collector’s problem, the point processes given by the times of $r^{th}$ arrivals for coupons of each type, centered and normalized in a proper way, converge toward a non-homogeneous Poisson point process. This result is then used to derive some generalizations and infinite-dimensional extensions of classical limit theorems on the topic.
</p>projecteuclid.org/euclid.ecp/1568253713_20191011220121Fri, 11 Oct 2019 22:01 EDTLocal nondeterminism and the exact modulus of continuity for stochastic wave equationhttps://projecteuclid.org/euclid.ecp/1568253714<strong>Cheuk Yin Lee</strong>, <strong>Yimin Xiao</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 8 pp..</p><p><strong>Abstract:</strong><br/>
We consider the linear stochastic wave equation driven by a Gaussian noise which is white in time and colored in space. We show that the solution satisfies a certain form of strong local nondeterminism and we use this property to derive the exact uniform modulus of continuity for the solution.
</p>projecteuclid.org/euclid.ecp/1568253714_20191011220121Fri, 11 Oct 2019 22:01 EDTHigh minima of non-smooth Gaussian processeshttps://projecteuclid.org/euclid.ecp/1568253715<strong>Zhixin Wu</strong>, <strong>Arijit Chakrabarty</strong>, <strong>Gennady Samorodnitsky</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this short note we study the asymptotic behaviour of the minima over compact intervals of Gaussian processes, whose paths are not necessarily smooth. We show that, beyond the logarithmic large deviation Gaussian estimates, this problem is closely related to the classical small-ball problem. Under certain conditions we estimate the term describing the correction to the large deviation behaviour. In addition, the asymptotic distribution of the location of the minimum, conditionally on the minimum exceeding a high threshold, is also studied.
</p>projecteuclid.org/euclid.ecp/1568253715_20191011220121Fri, 11 Oct 2019 22:01 EDTNew insights on concentration inequalities for self-normalized martingaleshttps://projecteuclid.org/euclid.ecp/1570845629<strong>Bernard Bercu</strong>, <strong>Taieb Touati</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We propose new concentration inequalities for self-normalized martingales. The main idea is to introduce a suitable weighted sum of the predictable quadratic variation and the total quadratic variation of the martingale. It offers much more flexibility and allows us to improve previous concentration inequalities. Statistical applications on autoregressive process, internal diffusion-limited aggregation process, and online statistical learning are also provided.
</p>projecteuclid.org/euclid.ecp/1570845629_20191011220121Fri, 11 Oct 2019 22:01 EDT