arXiv subjects

Martin Raič

Publications and source records attributed to Martin Raič.

5 recordsLinked to original sources

arXiv2025

A completion of counterexamples to the classical central limit theorem for pair- and triplewise independent and identically distributed random variables

By the Lindeberg-Lévy central limit theorem, standardized partial sums of a sequence of mutually independent and identically distributed random variables converge in law to the standard normal distribution. It is known that mutual independence cannot be relaxed to pairwise and even not triplewise independence. Counterexamples have been constructed for most marginal distributions: a recent construction works under a condition which excludes certain probability distributions with atomic parts, in particular discrete distributions in the `general position.' In the present paper, we show that this condition can be lifted: for any probability distribution $ F $ on the real line, which has finite variance and is not concentrated in a single point, there exists a sequence of triplewise independent random variables with distribution $ F $, such that its standardized partial sums converge in law to a distribution which is not normal. There is also scope for extension to $ k $-tuplewise independence.

math.PR↗

arXiv2023

Stein's density method for multivariate continuous distributions

This paper provides a general framework for Stein's density method for multivariate continuous distributions. The approach associates to any probability density function a canonical operator and Stein class, as well as an infinite collection of operators and classes which we call standardizations. These in turn spawn an entire family of Stein identities and characterizations for any continuous distribution on $\mathbb{R}^d$, among which we highlight those based on the score function and the Stein kernel. A feature of these operators is that they do not depend on normalizing constants. A new definition of Stein kernel is introduced and examined; integral formulas are obtained through a connection with mass transport, as well as ready-to-use explicit formulas for elliptical distributions. The flexibility of the kernels is used to compare in Stein discrepancy (and therefore 2-Wasserstein distance) between two normal distributions, Student and normal distributions, as well as two normal-gamma distributions. Upper and lower bounds on the 1-Wasserstein distance between continuous distributions are provided, and computed for a variety of examples: comparison between different normal distributions (improving on existing bounds in some regimes), posterior distributions with different priors in a Bayesian setting (including logistic regression), centred Azzalini--Dalla Valle distributions. Finally the notion of weak Stein equation and weak Stein factors is introduced. Bounds for solutions of the weak Stein equation are obtained for Lipschitz test functions if the distribution admits a Poincaré constant. We use these bounds to compare different copulas on the unit square in 1-Wasserstein distance.

math.PR↗

arXiv2020

On the error bound in the normal approximation for Jack measures

In this paper, we obtain uniform and non-uniform bounds on the Kolmogorov distance in the normal approximation for Jack deformations of the character ratio, by using Stein's method and zero-bias couplings. Our uniform bound comes very close to that conjectured by Fulman [J. Combin. Theory Ser. A, 108 (2004), 275--296]. As a by-product of the proof of the non-uniform bound, we obtain a Rosenthal-type inequality for zero-bias couplings.

math.PR↗

arXiv2019

A multivariate Berry--Esseen theorem with explicit constants

We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein's method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.

math.PR↗

arXiv2019

A multivariate central limit theorem for Lipschitz and smooth test functions

We provide an abstract multivariate central limit theorem with the Lindeberg-type error bounded in terms of Lipschitz functions (Wasserstein 1-distance) or functions with bounded second or third derivatives. The result is proved by means of Stein's method. For sums of i.i.d. random vectors with finite third absolute moment, the optimal rate of convergence is established (that is, we eliminate the logarithmic factor in the case of Lipschitz test functions). We indicate how the result could be applied to certain other dependence structures, but do not derive bounds explicitly.

math.PR↗