# On a Multivariate Strong Renewal Theorem

## 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.

## Keywords

Renewal Regular variation Infinitely divisible Large deviations## Mathematics Subject Classification (2010)

60K05 60F10## Notes

### Acknowledgements

The author would like to thank two referees and the AE for their careful reading of the paper and useful suggestions.

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