Stretched random walks and the behaviour of their summands

TL;DR

This paper investigates the behavior of individual summands in a random walk as its length grows and mean diverges, revealing they share the same value and affecting the walk's local path structure.

Contribution

It extends previous large deviation results by showing summands converge to a common value and analyzes the local path behavior under large deviation conditioning.

Findings

Summands tend to share the same value as the mean diverges.

Sample paths develop increasingly large and steep linear segments.

Results generalize large exceedance behavior in i.i.d. sums.

Abstract

This paper explores the joint behaviour of the summands of a random walk when their mean value goes to infinity as its length increases. It is proved that all the summands must share the same value, which extends previous results in the context of large exceedances of finite sums of i.i.d. random variables. Some consequences are drawn pertaining to the local behaviour of a random walk conditioned on a large deviation constraint on its end value. It is shown that the sample paths exhibit local oblic segments with increasing size and slope as the length of the random walk increases.