Notes on Random Walks in the Cauchy Domain of Attraction

2017/06/24 by Quentin Berger, Berger, Quentin · 3 citations
Mathematics · #60F05 #60F10 #60G50 #Analytic Number Theory Research #Benford’s Law and Fraud Detection #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications

paper · doi:10.48550/arxiv.1706.07924

openalex publication_date 2017/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has been in many cases left aside because of its additional technical difficulties. We prove here several results in that case: a Fuk-Nagaev inequality and a local version of it ; a large deviation theorem ; two types of local large deviation theorems. We also derive two important applications of these results: a sharp estimate of the tail of the first ladder epochs, and renewal theorems -- extending standard renewal theorems to the case of random walks. Most of our techniques carry through to the case of random walks in the domain of attraction of an α-stable law with α∈(0,2), so we also present results in that case, since some of them seem to be missing in the literature.

Citations

• Singularity Analysis of Generating Functions

Cited by

• The Random Walk Pinning Model I: Lower bounds on the free energy and disorder irrelevance
• Error Rates for Large Deviations in the domain of an α=1 stable law
• Sharp asymptotics for N-point correlation functions of coalescing heavy-tailed random walk

Related

• Strong renewal theorems and local large deviations for multivariate random walks and renewals
• The strong renewal theorem
• Random walks are determined by their trace on the positive half-line
• Local limit theorems for ladder epochs
• Local probabilities for random walks conditioned to stay positive