Weighted dependency graphs

TL;DR

This paper extends the theory of dependency graphs to weighted dependency graphs, providing new normality criteria and tools, with applications to various probabilistic models including permutations, random graphs, and Markov chains.

Contribution

It introduces weighted dependency graphs and offers criteria and tools for establishing normality in this broader context, with multiple applications.

Findings

Established normality criteria for weighted dependency graphs.

Applied the theory to random permutations, graphs, and Markov chains.

Extended existing theorems and answered open questions in the field.

Abstract

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context. We also provide generic tools to prove that some weighted graph is a weighted dependency graph for a given family of random variables. To illustrate the power of the theory, we give applications to the following objects: uniform random pair partitions, the random graph model $G(n,M)$, uniform random permutations, the symmetric simple exclusion process and multilinear statistics on Markov chains. The application to random permutations gives a bivariate extension of a functional central limit theorem of Janson and Barbour. On Markov chains, we answer positively an open question of Bourdon and Vall\'ee on the asymptotic normality of subword counts in random texts generated by a Markovian source.