On the generating functional of a convolution semigroup on a Hilbert-Lie group
On the Heyde theorem for discrete Abelian groups
Let X be a countable discrete Abelian group, Aut(X) the set of automorphisms of X, and I(X) the set of idempotent distributions on X. Assume that α₁, α₂, β₁, β₂ ∈ Aut(X) satisfy . Let ξ₁, ξ₂ be independent random variables with values in X and distributions μ₁, μ₂. We prove that the symmetry of the conditional distribution of L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ implies that μ₁, μ₂ ∈ I(X) if and only if the group X contains no elements of order two. This theorem can be considered as an analogue...
On the integrability of Banach space valued Walsh polynomials
On the Integral Representation in Convex Noncompact Sets of Tight Measures.
On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case
We consider the autoregressive model on ℝd defined by the stochastic recursion Xn = AnXn−1 + Bn, where {(Bn, An)} are i.i.d. random variables valued in ℝd× ℝ+. The critical case, when , was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measureν for the Markov chain {Xn}. In the present paper we prove that the weak limit of properly dilated measure ν exists and defines a homogeneous measure on ℝd ∖ {0}.
On the Itô formula in a Banach space.
On the Levy-Chintschin formula on locally compact maximally almost periodic groups.
On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution
We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...
On the local limit theorem for general lattice distribution.
On the mean speed of convergence of empirical and occupation measures in Wasserstein distance
In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably...
On the moments of sums of independent identically distributed random variables.
On the multiplicative ergodic theorem for uniquely ergodic systems
On the negative dependence in Hilbert spaces with applications
This paper introduces the notion of pairwise and coordinatewise negative dependence for random vectors in Hilbert spaces. Besides giving some classical inequalities, almost sure convergence and complete convergence theorems are established. Some limit theorems are extended to pairwise and coordinatewise negatively dependent random vectors taking values in Hilbert spaces. An illustrative example is also provided.
On the probability that two elements of a finite semigroup have the same right matrix
We study the probability that two elements which are selected at random with replacement from a finite semigroup have the same right matrix.
On the random ergodic theorem
On the rate of convergence of bootstrapped means in a Banach space.
On the reduction of a random basis
For p ≤ n, let b1(n),...,bp(n) be independent random vectors in with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios...
On the relation between Gaussian measures and convolution semigroups of local type.
On the Rényi dimension
On the sample paths of diagonal brownian motions on the infinite dimensional torus