Extremal independence in discrete random systems

2021/05/11 by Mikhail Isaev, I. V. Rodionov, Isaev, Mikhail +5
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2105.04917

openalex publication_date 2021/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X(n) ∈ ℝ^d be a sequence of random vectors, where n∈ℕ and d = d(n). Under certain weakly dependence conditions, we prove that the distribution of the maximal component of 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.

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