Extremal independence in discrete random systems
Mikhail Isaev, Igor Rodionov, Rui-Ray Zhang, Maksim Zhukovskii
TL;DR
This work studies extremal independence for high-dimensional, dependent random vectors by establishing conditions under which the maximum behaves as if the components were independent.The authors introduce a graph-based bound that separates mixing from declustering effects, and two bridging lemmas that transfer extremal behavior to approximating sequences, enabling broad applicability beyond i.i.d. settings.They obtain sharp upper and lower bounds for the probability of no exceedances and apply these results to Gaussian vectors and to extremal characteristics in random graphs and hypergraphs, yielding Gumbel limits for maxima of degrees, codegrees, and clique-extension counts.The results generalize classical extremal theory (including Berman-type conditions) to non-stationary and inhomogeneous dependent structures, providing a versatile framework for deriving asymptotic distributions in discrete stochastic networks.
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.