Random Walks in Cones: the Case of Nonzero Drift

TL;DR

This paper analyzes multidimensional random walks with nonzero drift confined within cones, deriving exit time asymptotics, studying conditioned process convergence, and constructing a version conditioned to stay forever.

Contribution

It provides new asymptotic results for exit times and constructs a conditioned process for multidimensional random walks with drift in cones.

Findings

Derived asymptotics for exit times from cones.

Established weak convergence of conditioned processes.

Constructed a process conditioned to never leave the cone.

Abstract

We consider multidimensional discrete valued random walks with nonzero drift killed when leaving general cones of the euclidian space. We find the asymptotics for the exit time from the cone and study weak convergence of the process conditioned on not leaving the cone. We get quasistationarity of its limiting distribution. Finally we construct a version of the random walk conditioned to never leave the cone.