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

TL;DR

This paper introduces a new approach using monotone couplings to derive sharper concentration inequalities for complex stochastic structures like graphs, trees, and branching processes, improving upon existing bounds.

Contribution

It develops a novel method leveraging monotone couplings from fixed point equations to enhance tail bounds in probabilistic models involving graphs and trees.

Findings

Sharper tail bounds for preferential attachment graphs

Improved concentration inequalities for branching processes

Enhanced bounds for random walk local times

Abstract

Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Pek\"oz, R\"ollin, and Ross (2016, arXiv:1309.4183 [math.PR]) 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.