Level-crossings of symmetric random walks and their application

TL;DR

This paper investigates the expected number of level-crossings in symmetric random walks, establishing inequalities and conditions for equality, with applications demonstrated in queuing theory.

Contribution

It introduces new inequalities for the expected level-crossings and identifies sequences where equality holds, extending understanding of random walk crossing behavior.

Findings

Established inequalities for expected level-crossings.

Proved existence of sequences with specific crossing expectations.

Applied results to queuing theory examples.

Abstract

Let $X_1$, $X_2$, $...$ be a sequence of independently and identically distributed random variables with $\mathsf{E}X_1=0$, and let $S_0=0$ and $S_t=S_{t-1}+X_t$, $t=1,2,...$, be a random walk. Denote $\tau={cases}\inf\{t>1: S_t\leq0\}, &\text{if} \ X_1>0, 1, &\text{otherwise}. {cases}$ Let $\alpha$ denote a positive number, and let $L_\alpha$ denote the number of level-crossings from the below (or above) across the level $\alpha$ during the interval $[0, \tau]$. Under quite general assumption, an inequality for the expected number of level-crossings is established. Under some special assumptions, it is proved that there exists an infinitely increasing sequence $\alpha_n$ such that the equality $\mathsf{E}L_{\alpha_n}=c\mathsf{P}\{X_1>0\}$ is satisfied, where $c$ is a specified constant that does not depend on $n$. The result is illustrated for a number of special random walks. We also give non-trivial examples from queuing theory where the results of this theory are applied.