Sojourn time in $\mathbb{Z}^+$ for the Bernoulli random walk on $\mathbb{Z}$

TL;DR

This paper investigates the distribution of the time a Bernoulli random walk spends in non-negative integers, providing simplified expressions for its probability distribution by modifying the sojourn time counting process.

Contribution

It introduces a modified sojourn time for Bernoulli walks that yields simpler probability distribution representations, extending previous symmetric case results.

Findings

Derived simpler formulas for the sojourn time distribution

Extended results beyond the symmetric case p=q=1/2

Connected discrete Bernoulli walk results to Le9vy's arcsine law

Abstract

Let $(S_k)_{k\ge 1}$ be the classical Bernoulli random walk on the integer line with jump parameters $p\in(0,1)$ and $q=1-p$. The probability distribution of the sojourn time of the walk in the set of non-negative integers up to a fixed time is well-known, but its expression is not simple. By modifying slightly this sojourn time--through a particular counting process of the zeros of the walk as done by Chung & Feller ["On fluctuations in coin-tossings", Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 605-608]-, simpler representations may be obtained for its probability distribution. In the aforementioned article, only the symmetric case ($p=q=1/2$) is considered. This is the discrete counterpart to the famous Paul L\'evy's arcsine law for Brownian motion.