Concentration inequalities from monotone couplings for graphs, walks, trees and branching processes

Tobias Johnson and Erol Peköz

Stochastic Processes and their Applications, 2022, vol. 152, issue C, 1-31

Abstract: Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Peköz et al. (2016) exploited characterizing distributional fixed point equations to obtain uniform error bounds for generalized gamma approximations using Stein’s method. Here we show how monotone couplings arising with these fixed point equations can be used to obtain sharper tail bounds that, in many cases, outperform competing moment-based bounds and the uniform bounds obtainable with Stein’s method. Applications are given to concentration inequalities for preferential attachment random graphs, branching processes, random walk local time statistics and the size of random subtrees of uniformly random binary rooted plane trees.

Keywords: Concentration inequality; Tail bound; Stein’s method; Preferential attachment graph; Galton–watson process (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spa.2022.06.012

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