Related papers: The universal Glivenko-Cantelli property

We generalize some classical results about quasicontinuous and separately continuous functions with values in metrizable spaces to functions with values in certain generalized metric spaces, called Maslyuchenko spaces. We establish…

Consider random matrices $A$, of dimension $m\times (m+n)$, drawn from an ensemble with probability density $f(\rmtr AA^\dagger)$, with $f(x)$ a given appropriate function. Break $A = (B,X)$ into an $m\times m$ block $B$ and the…

Denote by $\Gamma$ the set of pointwise good sequences. Those are sequences of real numbers $(a_k)$ such that for any measure preserving flow $(U_t)_{t\in \mathbb R}$ on a probability space and for any $f\in L^\infty$, the averages…

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions…

We study the class of selfless C*-probability spaces introduced by Robert. It is known that a selfless tracial algebra has strict comparison and a unique trace. We prove that for separable tracial C*-algebras, selflessness is equivalent to…

For continuous maps on a compact manifold M, particularly for those that do not preserve the Lebesgue measure m, we define the observable invariant probability measures as a generalization of the physical measures. We prove that any…

Uniformly finite homology is a coarse invariant for metric spaces; in particular, it is a quasi-isometry invariant for finitely generated groups. In this article, we study uniformly finite homology of finitely generated amenable groups and…

Let $G$ be a locally compact group and $\mu$ be a probability measure on $G$. We consider the convolution operator $\lambda_1(\mu)\colon L_1(G)\to L_1(G)$ given by $\lambda_1(\mu)f=\mu \ast f$ and its restriction $\lambda_1^0(\mu)$ to the…

We calculate the almost sure Hausdorff dimension of uniformly random self-similar fractals. These random fractals are generated from a finite family of similarities, where the linear parts of the mappings are independent uniformly…

Let $X$ be a centered random variable with unit variance, zero third moment, and such that $E[X^4] \ge 3$. Let $\{F_n : n\geq 1\}$ denote a normalized sequence of homogeneous sums of fixed degree $d\geq 2$, built from independent copies of…

We prove a.s. (almost sure) unisolvency of interpolation by continuous random sampling with respect to any given density, in spaces of multivariate a.e. (almost everywhere) analytic functions. Examples are given concerning polynomial and…

In 1964 R.Gangolli published a L\'{e}vy-Khintchine type formula which characterised $K$ bi-invariant infinitely divisible probability measures on a symmetric space $G/K$. His main tool was Harish-Chandra's spherical functions which he used…

Given a connected Lipschitz domain U we let L(U) be the subset of functions in 2nd order Sobolev space whose gradient (in the sense of trace) is equal to the inward pointing unit normal to U. The the Aviles Giga functional over L(U) serves…

In this paper, we identify a class of absolutely continuous probability distributions, and show that the differential entropy is uniformly convergent over this space under the metric of total variation distance. One of the advantages of…

This paper makes 3 contributions. First, it generalizes the Lindeberg\textendash Feller and Lyapunov Central Limit Theorems to Hilbert Spaces by way of $L^2$. Second, it generalizes these results to spaces in which sample failure and…

We show that for any isotropic log-concave probability measure $\mu$ on $\mathbb R^n$, for every $\varepsilon > 0$, every $1 \leq k \leq \sqrt{n}$ and any $E \in G_{n,k}$ there exists $F \in G_{n,k}$ with $d(E,F) < \varepsilon$ and…

We present a measure-theoretic condition for a property to hold ``almost everywhere'' on an infinite-dimensional vector space, with particular emphasis on function spaces such as $C^k$ and $L^p$. Like the concept of ``Lebesgue almost…

In this paper, generalized metrics mean metrics taking values in general linearly ordered Abelian groups. Using the Hahn fields, we first prove that for every generalized metric space, if the set of the Archimedean equivalence classes of…

We give estimates of the distance between the densities of the laws of two functionals $F$ and $G$ on the Wiener space in terms of the Malliavin-Sobolev norm of $F-G.$ We actually consider a more general framework which allows one to treat…

Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure, of any kind, there is a structure in $\mathcal{C}$ with the same…