Extremal independence in discrete random systems

TL;DR

This paper establishes conditions under which the maximum components of dependent random vectors behave similarly to independent ones, providing new bounds and applications to random hypergraphs and graphs.

Contribution

It introduces new bounds for extremal probabilities and extends extremal independence results to Gaussian vectors and discrete structures.

Findings

Asymptotic equivalence of maxima distributions under weak dependence

New bounds for probabilities of no occurrence of events

Applications to extremal characteristics in random hypergraphs and graphs

Abstract

Let $\mathbf{X}(n) \in \mathbb{R}^d$ be a sequence of random vectors, where $n\in\mathbb{N}$ and $d = d(n)$. Under certain weakly dependence conditions, we prove that the distribution of the maximal component of $\mathbf{X}$ and the distribution of the maximum of their independent copies are asymptotically equivalent. Our result on extremal independence relies on new lower and upper bounds for the probability that none of a given finite set of events occurs. As applications, we obtain the distribution of various extremal characteristics of random discrete structures such as maximum codegree in binomial random hypergraphs and the maximum number of cliques sharing a given vertex in binomial random graphs. We also generalise Berman-type conditions for a sequence of Gaussian random vectors to possess the extremal independence property.