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.
Mots-clés : random walk, sojourn time, generating function
@article{PS_2012__16__324_0, 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}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps/2010013/} }
TY - JOUR AU - Lachal, Aimé TI - Sojourn time in $\mathbb {Z}^{+}$ for the Bernoulli random walk on $\mathbb {Z}$ JO - ESAIM: Probability and Statistics PY - 2012 SP - 324 EP - 351 VL - 16 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2010013/ DO - 10.1051/ps/2010013 LA - en ID - PS_2012__16__324_0 ER -
%0 Journal Article %A Lachal, Aimé %T Sojourn time in $\mathbb {Z}^{+}$ for the Bernoulli random walk on $\mathbb {Z}$ %J ESAIM: Probability and Statistics %D 2012 %P 324-351 %V 16 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2010013/ %R 10.1051/ps/2010013 %G en %F PS_2012__16__324_0
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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