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Let $(S_k)_{k\ge 1}$ be the classical Bernoulli random walk on the integerline with jump parameters $p\in(0,1)$ and $q=1-p$. The probability distributionof the sojourn time of the walk in the set of non-negative integers up to afixed time is well-known, but its expression is not simple. By modifyingslightly this sojourn time--through a particular counting process of the zerosof 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 maybe obtained for its probability distribution. In the aforementioned article,only the symmetric case ($p=q=1/2$) is considered. This is the discretecounterpart to the famous Paul L\'evy's arcsine law for Brownian motion.

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