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

Abstract

Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Peköz, Röllin, 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.

arXiv comment30 pages; minor changes in response to referees' comments

References in corpus (9)

• Emergence of scaling in random networks
• Stein's method and the zero bias transformation with application to simple random sampling
• New rates for exponential approximation and the theorems of Rényi and Yaglom
• A short survey of Stein's method
• Size biased couplings and the spectral gap for random regular graphs
• Generalized gamma approximation with rates for urns, walks and trees
• On the second eigenvalue of random bipartite biregular graphs
• Relaxation of monotone coupling conditions: Poisson approximation and beyond
• Particle density in diffusion-limited annihilating systems