On Multivariate Strong Renewal Theorem

TL;DR

This paper develops a probabilistic approach to the multivariate Strong Renewal Theorem, accommodating mixed lattice and nonlattice distributions, and provides weaker assumptions for various applications.

Contribution

It introduces a version of the SRT for multivariate distributions with mixed lattice structures and derives bounds to control small-n contributions, broadening applicability.

Findings

Established a version of the SRT for mixed lattice-nonlattice distributions.

Derived bounds controlling the small-n contribution in the renewal process.

Provided applications with weaker assumptions, unifying and extending known results.

Abstract

This paper takes the so-called probabilistic approach to the Strong Renewal Theorem (SRT) for multivariate distributions in the domain of attraction of a stable law. A version of the SRT is obtained that allows any kind of lattice-nonlattice composition of a distribution. A general bound is derived to control the so-called "small-$n$ contribution", which arises from random walk paths that have a relatively small number of steps but make large cumulative moves. The asymptotic negligibility of the small-$n$ contribution is essential to the SRT. Applications of the SRT are given, including some that provide a unified treatment to known results but with substantially weaker assumptions.