The tail of the length of an excursion in a trap of random size

TL;DR

This paper analyzes the tail behavior of the excursion length in a random walk with a geometrically distributed trap size, revealing non-regular variation and explicit oscillations in the tail probability.

Contribution

It provides a detailed characterization of the tail decay and oscillations of the excursion length in a random walk with a random trap size, which was previously not well understood.

Findings

Tail probability decreases like a power law with exponent 

The tail is not regularly varying, exhibiting oscillations

Explicit formulas for the oscillations of the tail probability

Abstract

Consider a random walk with a drift to the right on $\{0,\ldots,k\}$ where $k$ is random and geometrically distributed. We show that the tail $P[T>t]$ of the length $T$ of an excursion from $0$ decreases up to constants like $t^{-\varrho}$ for some $\varrho>0$ but is not regularly varying. We compute the oscillations of $t^\varrho\,P[T>t]$ as $t\to\infty$ explicitly.