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We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…

Probability · Mathematics 2023-10-25 Aurelien Gribinski

Benford's law describes a common phenomenon among many naturally occurring data sets and distributions in which the leading digits of the data are distributed with the probability of a first digit of $d$ base $B$ being…

Probability · Mathematics 2019-10-30 Rebecca F. Durst , Steven J. Miller

In this paper, on the sublinear expectation space, we establish a comparison theorem between independent and convolutionary random vectors, which states that the partial sums of those two sequences of random vectors are identically…

Probability · Mathematics 2017-10-05 Ning Zhang , Yuting Lan

In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $\mu$ and $\nu$ such that $\nu$ is log-concave with respect to $\mu$.…

Probability · Mathematics 2022-10-24 Arturo Jaramillo , James Melbourne

We investigate conditions for the existence of the limiting conditional distribution of a bivariate random vector when one component becomes large. We revisit the existing literature on the topic, and present some new sufficient conditions.…

Probability · Mathematics 2010-02-21 Anne-Laure Fougères , Philippe Soulier

We present a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation. This approach allows us to…

Operator Algebras · Mathematics 2025-02-06 Octavio Arizmendi , Katsunori Fujie , Daniel Perales , Yuki Ueda

This paper addresses the following classical question: giving a sequence of identically distributed random variables in the domain of attraction of a normal law, does the associated linear process satisfy the central limit theorem? We study…

Probability · Mathematics 2011-06-01 Magda Peligrad , Hailin Sang

We consider the evolution by crystalline curvature of a planar set in a stratified medium, modeled by a periodic forcing term. We characterize the limit evolution law as the period of the oscillations tends to zero. Even if the model is…

Analysis of PDEs · Mathematics 2017-07-13 Andrea Braides , Annalisa Malusa , Matteo Novaga

We study mixtures of free, monotone, and Boolean independence described by a directed graph $G = (V,E)$ in the context of $\mathcal{T}$-free convolutions of Jekel and Liu. We prove general limit theorems for the associated additive…

Probability · Mathematics 2025-07-31 David Jekel , Lahcen Oussi , Janusz Wysoczański

Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (i) the limit distribution is…

Probability · Mathematics 2013-05-14 Ivan Nourdin , Guillaume Poly

We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the $*$-distribution of the product of two selfadjoint freely independent random variables,…

Operator Algebras · Mathematics 2020-09-24 Maxime Fevrier , Mitja Mastnak , Alexandru Nica , Kamil Szpojankowski

We study the asymptotic behaviour of a random walk whose evolution is dependent on the state of an itself dynamically evolving environment. In particular, we extend our previous results in [Bethuelsen and V\"ollering, 2016] and prove a…

Probability · Mathematics 2024-11-21 Stein Andreas Bethuelsen , Florian Völlering

We establish a central limit theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb{N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the…

Probability · Mathematics 2024-05-01 Cécilia Lancien , Patrick Oliveira Santos , Pierre Youssef

We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^1([0,1]), and discuss generalizations to discrete random variables. A consequence…

Probability · Mathematics 2010-09-15 Steven J. Miller , Mark J. Nigrini

Well known biological approximations are universal, i.e. invariant to transformations from one species to another. With no other experimental data, such invariance yields exact conservation (with respect to biological diversity and…

Statistical Mechanics · Physics 2007-05-23 Mark Ya. Azbel'

This paper investigates asymptotic properties of multifractal products of random fields. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces $L_q.$ New results on the rate of…

Probability · Mathematics 2022-02-08 Illia Donhauzer , Andriy Olenko

Conditioned limit laws constitute an important and well developed framework of extreme value theory that describe a broad range of extremal dependence forms including asymptotic independence. We explore the assumption of conditional…

Probability · Mathematics 2015-12-31 Ioannis Papastathopoulos

We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on [0,\infty), called the free regular measures. We…

Probability · Mathematics 2012-12-20 Octavio Arizmendi , Takahiro Hasebe , Noriyoshi Sakuma

We introduce the notion of operator-valued infinitesimal (OVI) independence for the Boolean and monotone cases. Then show that OVI Boolean (resp. monotone) independence is equivalent to the operator-valued Boolean (resp. monotone)…

Operator Algebras · Mathematics 2021-08-27 Daniel Perales , Pei-Lun Tseng

We show that the laws of autocorrelations decay in texts are closely related to applicability limits of language models. Using distributional semantics we empirically demonstrate that autocorrelations of words in texts decay according to a…

Computation and Language · Computer Science 2023-05-12 Nikolay Mikhaylovskiy , Ilya Churilov