Ordered random walks with heavy tails

TL;DR

This paper extends the study of random walks in Weyl chambers by analyzing heavy-tailed increments with regular variation, revealing different asymptotic behaviors and constructing conditioned processes under these new conditions.

Contribution

It introduces the analysis of heavy-tailed increments in Weyl chamber random walks, showing altered asymptotics and constructing conditioned processes for this case.

Findings

Asymptotic behavior of exit times differs with heavy tails

Constructed conditioned processes on a partial compactification

Identified the impact of tail index on walk behavior

Abstract

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $\alpha<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using thisinformation, construct a conditioned process which lives on a partial compactification of the Weyl chamber.