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

ESAIM: Probability and Statistics (2012)

- Volume: 16, page 324-351
- ISSN: 1292-8100

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topLachal, Aimé. "Sojourn time in ℤ+ for the Bernoulli random walk on ℤ." ESAIM: Probability and Statistics 16 (2012): 324-351. <http://eudml.org/doc/274397>.

@article{Lachal2012,

abstract = {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.},

author = {Lachal, Aimé},

journal = {ESAIM: Probability and Statistics},

keywords = {random walk; sojourn time; generating function},

language = {eng},

pages = {324-351},

publisher = {EDP-Sciences},

title = {Sojourn time in ℤ+ for the Bernoulli random walk on ℤ},

url = {http://eudml.org/doc/274397},

volume = {16},

year = {2012},

}

TY - JOUR

AU - Lachal, Aimé

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

JO - ESAIM: Probability and Statistics

PY - 2012

PB - EDP-Sciences

VL - 16

SP - 324

EP - 351

AB - 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.

LA - eng

KW - random walk; sojourn time; generating function

UR - http://eudml.org/doc/274397

ER -

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