Persistence of heavy-tailed sample averages: principle of infinitely many big jumps

Abstract

We consider the sample average of a centered random walk in ${\mathbb{R}}^d$ with regularly varying step size distribution. For the first exit time from a compact convex set A not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.