The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem

TL;DR

This paper derives the precise asymptotic probabilities for a multivariate random walk to hit a translated positive orthant, extending previous two-dimensional results to higher dimensions with complex boundary geometries.

Contribution

It introduces new techniques combining large deviation methods and auxiliary constructions to handle boundary corners in multivariate boundary crossing problems.

Findings

Exact asymptotics for hitting probabilities in multivariate random walks.

Extension of previous results to boundaries with corners.

Methodology applicable to more general target sets.

Abstract

For a multivariate random walk with i.i.d. jumps satisfying the Cramer moment condition and having a mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results from a paper by F. Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a recent series of papers by A. Borovkov and A. Mogulskii with new auxiliary constructions, which enable us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a "corner" at the "most probable hitting point". We also discuss how our results can be extended to the case of more general target sets.