On a Multivariate Strong Renewal Theorem

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.