We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$, the relative frequencies of sets $C\in\mathcal{C}$ converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of $\mathcal{C}$. The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.

We prove that when a sequence of Lévy processes X⁽ⁿ⁾ or a normed sequence of random walks S⁽ⁿ⁾ converges a.s. on the Skorokhod space toward a Lévy process X, the sequence L⁽ⁿ⁾ of local times at the supremum of X⁽ⁿ⁾ converges uniformly on compact sets in probability toward the local time at the supremum of X. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law toward the ladder processes of X. As an application, we show that in general, the sequence S⁽ⁿ⁾ conditioned to stay positive converges weakly, jointly with its local time at the future minimum, toward the corresponding functional for the limiting process X. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.

We call a right-continuous increasing process K_x a partial right inverse (PRI) of a given Lévy process X if X_Kx = x for at least all x in some random interval [0, ζ) of positive length. In this paper, we give a necessary and sufficient condition for the existence of a PRI in terms of the Lévy triplet.

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-* ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-* mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices.

The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.

In this paper, we study the superconvergence phenomenon in the free central limit theorem for identically distributed, unbounded summands. We prove not only the uniform convergence of the densities to the semicircular density but also their L^p-convergence to the same limit for p > 1/2. Moreover, an entropic central limit theorem is obtained as a consequence of the above results.

We prove the central limit theorem for the volume and the f-vector of the Poisson random polytope Π_η in a fixed convex polytope P ⊂ ℝ^d. Here, Π_η is the convex hull of the intersection of a Poisson process X of intensity η with P.

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao’s divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized Itô formula.

We give an effective upper escape rate function for Brownian motion on a complete Riemannian manifold in terms of the volume growth of the manifold. An important step in the work is estimating the small tail probability of the crossing time between two concentric geodesic spheres by reflecting Brownian motions on the larger geodesic ball.

We consider long-range percolation on ℤ^d, where the probability that two vertices at distance r are connected by an edge is given by p(r) = 1 − exp[−λ(r)] ∈ (0, 1) and the presence or absence of different edges are independent. Here, λ(r) is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the k-ball around the origin, $|\mathcal{B}_{k}|$, that is, the number of vertices that are within graph-distance k of the origin, for k → ∞, for different λ(r). We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying λ(r) exist, for which, respectively:

• $|\mathcal{B}_{k}|^{1/k}\to\infty$> almost surely;

• there exist 1 < a₁ < a₂ < ∞ such that $\lim_{k\to \infty}\mathbb{P}(a_{1}< |\mathcal{B}_{k}|^{1/k}< a_{2})=1$;

• $|\mathcal{B}_{k}|^{1/k}\to1$ almost surely.

This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, R₀, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.

For parameters p ∈ [0, 1] and q > 0 such that the Fortuin–Kasteleyn (FK) random-cluster measure Φ_p,q^ℤd for ℤ^d with parameters p and q is unique, the q-divide and color [DaC(q)] model on ℤ^d is defined as follows. First, we draw a bond configuration with distribution Φ_p,q^ℤd. Then, to each (FK) cluster (i.e., to every vertex in the FK cluster), independently for different FK clusters, we assign a spin value from the set {1, 2, …, s} in such a way that spin i has probability a_i.

In this paper, we prove that the resulting measure on spin configurations is a Gibbs measure for small values of p and is not a Gibbs measure for large p, except in the special case of q ∈ {2, 3, …}, a₁ = a₂ = ⋯ = a_s = 1/q, when the DaC(q) model coincides with the q-state Potts model.

We prove the almost sure invariance principle for stationary ℝ^d-valued random processes (with very precise dimension-independent error terms), solely under a strong assumption concerning the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains using strong or weak spectral perturbation arguments.

Let X₁, …, X_n be independent with zero means, finite variances σ₁², …, σ_n² and finite absolute third moments. Let F_n be the distribution function of (X₁ + ⋯ + X_n)/σ, where σ² = ∑_i=1ⁿσ_i², and Φ that of the standard normal. The L¹-distance between F_n and Φ then satisfies $$\Vert F_{n}-\Phi\Vert_{1}\le\frac{1}{\sigma^{3}}\sum_{i=1}^{n}E|X_{i}|^{3}.$$

In particular, when X₁, …, X_n are identically distributed with variance σ², we have $$\Vert F_{n}-\Phi\Vert_{1}\le\frac{E|X_{1}|^{3}}{\sigma^{3}\sqrt{n}} \text{for all } n ∈ ℕ,$$ corresponding to an L¹-Berry–Esseen constant of 1.

Consider the mutually catalytic branching process with finite branching rate γ. We show that as γ → ∞, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process.

This is the first paper in a series or three, wherein we also construct an interacting version of this process and study its long-time behavior.