Related papers: Limit theorems for additive c-free convolution

We prove an analogue of the portmanteau theorem on weak convergence of probability measures allowing measures which are unbounded on an underlying metric space but finite on the complement of any Borel neighbourhood of a fixed element.

In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are…

We extend to arbitrary measures results of Bao, Erd\"os, Schnelli, Moreillon, and Ji on the connectedness of the supports of additive convolutions of measures on \mathbb{R} and of free multiplicative convolutions of measures on…

A convergence theorem for martingales with c\`adl\`ag trajectories (right continuous with left limits everywhere) is obtained in the sense of the weak dual topology on Hilbert space, under conditions that are much weaker than those required…

We give a streamlined proof of the limit theorems for the free additive convolution of infinitesimal triangular arrays of probability measures on the real line. The result was first proved by Chistyakov and G\"otze using analytic…

Filter convergence of vector lattice-valued measures is considered, in order to deduce theorems of convergence for their decompositions. First the $\sigma$-additive case is studied, without particular assumptions on the filter; later the…

Let $\{ A_k\}_{k=1}^\infty$ be a sequence of finite subsets of $\mathbb{R}^d$ satisfying that $\# A_k \ge 2$ for all integers $k \ge 1$. In this paper, we first give a sufficient and necessary condition for the existence of the infinite…

Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical…

Minimizing divergence measures under a constraint is an important problem. We derive a sufficient condition that binary divergence measures provide lower bounds for symmetric divergence measures under a given triangular discrimination or…

The main subject of this paper is equilibrium problems on an unbounded conductor $\Sigma$ of the complex plane in the presence of a weakly admissible external field. An admissible external field $Q$ on $\Sigma$ satisfies, along with other…

This paper proves several weak limit theorems for the joint version of extreme order statistics and partial sums of independently and identically distributed random variables. The results are also extended to almost sure limit version.

We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically…

For different classes of measure preserving transformations, we investigate collections of sets that exhibit the property of lightly mixing. Lightly mixing is a stronger property than topological mixing, and requires that a lim inf is…

Rational weak mixing is a measure theoretic version of Krickeberg's strong ratio mixing property for infinite measure preserving transformations. It requires "{\tt density}" ratio convergence for every pair of measurable sets in a dense…

We give general conditions for the central limit theorem and weak convergence to Brownian motion (the weak invariance principle / functional central limit theorem) to hold for observables of compact group extensions of nonuniformly…

Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability…

In this paper we define the closure under weak convergence of the class of p-tempered {\alpha}-stable distributions. We give necessary and sufficient conditions for convergence of sequences in this class. Moreover, we show that any element…

The exact conditions on valid pointer states for weak measurements are derived. It is demonstrated that weak measurements can be performed with any pointer state with vanishing probability current density. This condition is found both for…

We show that a probability measure is not a nontrivial free additive convolution if it puts no mass in an interval whose endpoints are atoms. The analogous results for free multiplicative convolutions are proved as well. The proofs use…

In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions…