We show that the convex hull of a large i.i.d. sample from an absolutely continuous log-concave distribution approximates a predetermined convex body in the logarithmic Hausdorff distance and in the Banach–Mazur distance. For log-concave distributions that decay super-exponentially, we also have approximation in the Hausdorff distance. These results are multivariate versions of the Gnedenko law of large numbers, which guarantees concentration of the maximum and minimum in the one-dimensional case.

We provide quantitative bounds in terms of the number of points and the dimension of the ambient space.

We study the minimal sample size $N=N(n)$ that suffices to estimate the covariance matrix of an $n$-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. We establish the optimal bound $N=O(n)$ for every distribution whose $k$-dimensional marginals have uniformly bounded $2+\varepsilon$ moments outside the sphere of radius $O(\sqrt{k})$. In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan–Lovasz–Simonovits problem, which was recently solved by Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535–561]. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption—uniformly bounded $2+\varepsilon$ moments of one-dimensional marginals. Our argument consists of randomizing the spectral sparsifier, a deterministic tool developed recently by Batson, Spielman and Srivastava [SIAM J. Comput. 41 (2012) 1704–1721]. The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points.

We give a characterization of transport-entropy inequalities in metric spaces. As an application we deduce that such inequalities are stable under bounded perturbation (Holley–Stroock perturbation lemma).

We consider the problem of optimal incomplete transportation between the empirical measure on an i.i.d. uniform sample on the $d$-dimensional unit cube $[0,1]^{d}$ and the true measure. This is a family of problems lying in between classical optimal transportation and nearest neighbor problems. We show that the empirical cost of optimal incomplete transportation vanishes at rate $O_{P}(n^{-1/d})$, where $n$ denotes the sample size. In dimension $d\geq3$ the rate is the same as in classical optimal transportation, but in low dimension it is (much) higher than the classical rate.

We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties for the environment as seen from the position of the walker, that is, the environment process. We can transfer the rate of mixing in time of the environment to the rate of mixing of the environment process with a loss of at most polynomial order. Therefore the method is applicable to environments with sufficiently fast polynomial mixing. We obtain unique ergodicity of the environment process. Moreover, the unique invariant measure of the environment process depends continuously on the jump rates of the walker.

As a consequence we obtain the law of large numbers and a central limit theorem with nondegenerate variance for the position of the walk.

In this paper we study the Poisson process over a $\sigma$-finite measure-space equipped with a measure preserving transformation or a group of measure preserving transformations. For a measure-preserving transformation $T$ acting on a $\sigma$-finite measure-space $X$, the Poisson suspension of $T$ is the associated probability preserving transformation $T_{*}$ which acts on realization of the Poisson process over $X$. We prove ergodicity of the Poisson-product $T\times T_{*}$ under the assumption that $T$ is ergodic and conservative. We then show, assuming ergodicity of $T\times T_{*}$, that it is impossible to deterministically perform natural equivariant operations: thinning, allocation or matching. In contrast, there are well-known results in the literature demonstrating the existence of isometry equivariant thinning, matching and allocation of homogenous Poisson processes on $\mathbb{R}^{d}$. We also prove ergodicity of the “first return of left-most transformation” associated with a measure preserving transformation on $\mathbb{R}_{+}$, and discuss ergodicity of the Poisson-product of measure preserving group actions, and related spectral properties.

We consider an extension of the Monge–Kantorovitch optimal transportation problem. The mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and the diffusion coefficients of the continuous semimartingale. The optimal transportation problem minimizes the cost among all continuous semimartingales with given initial and terminal distributions. Our first main result is an extension of the Kantorovitch duality to this context. We also suggest a finite-difference scheme combined with the gradient projection algorithm to approximate the dual value. We prove the convergence of the scheme, and we derive a rate of convergence.

We finally provide an application in the context of financial mathematics, which originally motivated our extension of the Monge–Kantorovitch problem. Namely, we implement our scheme to approximate no-arbitrage bounds on the prices of exotic options given the implied volatility curve of some maturity.

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $\mu_{i}$ and $\nu_{i}$, $i=1,2$, are close to each other in terms of the Lé vy metric and if the free convolution $\mu_{1}\boxplus\mu_{2}$ is sufficiently smooth, then $\nu_{1}\boxplus\nu_{2}$ is absolutely continuous, and the densities of measures $\nu_{1}\boxplus\nu_{2}$ and $\mu_{1}\boxplus\mu_{2}$ are close to each other. In particular, convergence in distribution $\mu_{1}^{(n)}\rightarrow\mu_{1}$, $\mu_{2}^{(n)}\rightarrow\mu_{2}$ implies that the density of $\mu_{1}^{(n)}\boxplus\mu_{2}^{(n)}$ is defined for all sufficiently large $n$ and converges to the density of $\mu_{1}\boxplus\mu_{2}$. Some applications are provided, including: (i) a new proof of the local version of the free central limit theorem, and (ii) new local limit theorems for sums of free projections, for sums of $\boxplus$-stable random variables and for eigenvalues of a sum of two $N$-by-$N$ random matrices.

All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star–triangle transformation and the box-crossing property. The exponents in question are the one-arm exponent $\rho$, the $2j$-alternating-arms exponents $\rho_{2j}$ for $j\ge1$, the volume exponent $\delta$, and the connectivity exponent $\eta$. By earlier results of Kesten, this implies universality also for the near-critical exponents $\beta$, $\gamma$, $\nu$, $\Delta$ (assuming these exist) for any of these models that satisfy a certain additional hypothesis, such as the homogeneous bond percolation models on these three lattices.

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Székely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov’s fundamental result on $\mathbb{R}^{d}$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin–Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.

We consider a spin system obtained by coupling two distinct Sherrington–Kirkpatrick (SK) models with the same temperature and external field whose Hamiltonians are correlated. The disorder chaos conjecture for the SK model states that the overlap under the corresponding Gibbs measure is essentially concentrated at a single value. In the absence of external field, this statement was first confirmed by Chatterjee [Disorder chaos and multiple valleys in spin glasses (2009) Preprint]. In the present paper, using Guerra’s replica symmetry breaking bound, we prove that the SK model is also chaotic in the presence of the external field and the position of the overlap is determined by an equation related to Guerra’s bound and the Parisi measure.

We prove diffusive lower bounds on the rate of escape of the random walk on infinite transitive graphs. Similar estimates hold for finite graphs, up to the relaxation time of the walk. Our approach uses nonconstant equivariant harmonic mappings taking values in a Hilbert space. For the special case of discrete, amenable groups, we present a more explicit proof of the Mok–Korevaar–Schoen theorem on the existence of such harmonic maps by constructing them from the heat flow on a Følner set.

The main results in this paper are about the full coalescence time $\mathsf{C}$ of a system of coalescing random walks over a finite graph $G$. Letting $\mathsf{m}(G)$ denote the mean meeting time of two such walkers, we give sufficient conditions under which $\mathbf{E}[\mathsf{C}]\approx2\mathsf{m}(G)$ and $\mathsf{C}/\mathsf{m}(G)$ has approximately the same law as in the “mean field” setting of a large complete graph. One of our theorems is that mean field behavior occurs over all vertex-transitive graphs whose mixing times are much smaller than $\mathsf{m}(G)$; this nearly solves an open problem of Aldous and Fill and also generalizes results of Cox for discrete tori in $d\geq2$ dimensions. Other results apply to nonreversible walks and also generalize previous theorems of Durrett and Cooper et al. Slight extensions of these results apply to voter model consensus times, which are related to coalescing random walks via duality.

Our main proof ideas are a strengthening of the usual approximation of hitting times by exponential random variables, which give results for nonstationary initial states; and a new general set of conditions under which we can prove that the hitting time of a union of sets behaves like a minimum of independent exponentials. In particular, this will show that the first meeting time among $k$ random walkers has mean $\approx\mathsf{m}(G)/\bigl({\matrix{k\\2}}\bigr)$.

Consider a $d$-dimensional Brownian motion in a random potential defined by attaching a nonnegative and polynomially decaying potential around Poisson points. We introduce a repulsive interaction between the Brownian path and the Poisson points by weighting the measure by the Feynman–Kac functional. We show that under the weighted measure, the Brownian motion tends to localize around the origin. We also determine the scaling limit of the path and also the limit shape of the random potential.

It is, perhaps, surprising that the location of the unique supremum of a stationary process on an interval can fail to be uniformly distributed over that interval. We show that this distribution is absolutely continuous in the interior of the interval and describe very specific conditions the density has to satisfy. We establish universal upper bounds on the density and demonstrate their optimality.

We give short proofs of two classical results about the position of the extremal particle in a branching Brownian motion, one concerning the median position and another the almost sure behaviour.

It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_{n}(z)=\sum_{k=0}^{n}\xi_{k}z^{k}$ with i.i.d. coefficients $\xi_{0},\ldots,\xi_{n}$ concentrate a.s. near the unit circle as $n\to\infty$ if and only if ${\mathbb{E}\log_{+}}|\xi_{0}|<\infty$. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like $L({\log}|t|)({\log}|t|)^{-\alpha}$ as $t\to\infty$, where $\alpha\geq0$, and $L$ is a slowly varying function. Under this assumption, the structure of complex and real roots of $G_{n}$ is described in terms of the least concave majorant of the Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)}\,du\,dv$.

We consider nondegenerate, finitely supported random walks on a finitely generated Gromov hyperbolic group. We show that the entropy and the escape rate are Lipschitz functions of the probability if the support remains constant.

Let $X_{1},X_{2},\ldots$ be a centred sequence of weakly stationary random variables with spectral measure $F$ and partial sums $S_{n}=X_{1}+\cdots+X_{n}$. We show that $\operatorname{var} (S_{n})$ is regularly varying of index $\gamma$ at infinity, if and only if $G(x):=\int_{-x}^{x}F(\mathrm{d} x)$ is regularly varying of index $2-\gamma$ at the origin ($0<\gamma<2$).

We condition super-Brownian motion on “boundary statistics” of the exit measure $X_{D}$ from a bounded domain $D$. These are random variables defined on an auxiliary probability space generated by sampling from the exit measure $X_{D}$. Two particular examples are: conditioning on a Poisson random measure with intensity $\beta X_{D}$ and conditioning on $X_{D}$ itself. We find the conditional laws as $h$-transforms of the original SBM law using Dynkin’s formulation of $X$-harmonic functions. We give explicit expression for the (extended) $X$-harmonic functions considered. We also obtain explicit constructions of these conditional laws in terms of branching particle systems. For example, we give a fragmentation system description of the law of SBM conditioned on $X_{D}=\nu$, in terms of a particle system, called the backbone. Each particle in the backbone is labeled by a measure $\tilde{\nu}$, representing its descendants’ total contribution to the exit measure. The particle’s spatial motion is an $h$-transform of Brownian motion, where $h$ depends on $\tilde{\nu}$. At the particle’s death two new particles are born, and $\tilde{\nu}$ is passed to the newborns by fragmentation.

We prove a strong approximation result for the empirical process associated to a stationary sequence of real-valued random variables, under dependence conditions involving only indicators of half lines. This strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero, as soon as the correlations decrease more rapidly than $n^{-1-\delta}$ for some positive $\delta$. This shows that our conditions are in some sense optimal.