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.
Keywords: 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},
year = {2012},
publisher = {EDP-Sciences},
volume = {16},
doi = {10.1051/ps/2010013},
zbl = {1275.60046},
language = {en},
url = {http://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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
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