Integral criteria for Strong Renewal Theorems with infinite mean

2013/12/20 by Zhiyi Chi, Chi, Zhiyi
Mathematics · #60F10 #60K05 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F10 #msc:60K05

paper · pdf · doi:10.48550/arxiv.1312.6089

arxiv created 2014/04/15 · arxiv updated 2014/04/16

Abstract

Let F be a probability measure on ℝ in the domain of attraction of a stable law with exponent α∈ (0, 1). We establish integral criteria on F that significantly expand the probabilistic approach to Strong Renewal Theorems (SRTs). The criterion for α∈ (0,1/2] is much weaker than currently available ones and in some cases provides sufficient and necessary conditions for the SRT. The criterion for α∈ (1/2, 1) establishes the SRT in full generality and in a unified way, barring the Limit Local Theorems employed. As an application, for infinitely divisible F, an integral criterion on its Lévy measure is established for the SRT. As another application, for F in the domain of attraction of a stable law without centering, an integral criterion on F is established for the SRT for the ladder height process of a random walk with step distribution F.

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