Sojourn time in ℤ + for the Bernoulli random walk on ℤ
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 324-351.

Let (Sk)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. In the present paper, we write out a representation for this probability distribution in the general case together with others related to the random walk subject to a possible conditioning. The main tool is the use of generating functions.

DOI : 10.1051/ps/2010013
Classification : 60G50, 60J22, 60J10, 60E10
Mots clés : random walk, sojourn time, generating function 
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     author = {Lachal, Aim\'e},
     title = {Sojourn time in $\mathbb {Z}^{+}$ for the {Bernoulli} random walk on $\mathbb {Z}$},
     journal = {ESAIM: Probability and Statistics},
     pages = {324--351},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2010013},
     zbl = {1275.60046},
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     url = {http://archive.numdam.org/articles/10.1051/ps/2010013/}
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Lachal, Aimé. Sojourn time in $\mathbb {Z}^{+}$ for the Bernoulli random walk on $\mathbb {Z}$. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 324-351. doi : 10.1051/ps/2010013. http://archive.numdam.org/articles/10.1051/ps/2010013/

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