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

## TL;DR

This paper investigates the tail behavior of the first exit time for a heavy-tailed random walk, revealing it follows a lognormal distribution and that large exit times are typically caused by a logarithmic number of big jumps.

## Contribution

It establishes the lognormal tail distribution of the exit time and characterizes the typical jump pattern leading to large exit times for heavy-tailed random walks.

## Key findings

- First exit time tail is of lognormal type.
- Large exit times are caused by a logarithmic number of big jumps.
- The behavior is characterized for random walks with regularly varying step sizes.

## 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.