Sojourn time in ℤ⁺ for the Bernoulli random walk on ℤ

E-ISSN： 1262-3318|16|issue|324-351

ISSN： 1292-8100

Source： ESAIM: Probability and Statistics, Vol.16, Iss.issue, 2012-08, pp. : 324-351

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Abstract

Let (S_k)_k≥1 be the classical Bernoulli random walk on the integer line with jump parameters p ∈ (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 [Proc. Nat. Acad. Sci. USA 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 Lévy’s arcsine law for Brownian motion.