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

Aimé Lachal

ESAIM: Probability and Statistics (2012)

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

Abstract

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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.

How to cite

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Lachal, Aimé. "Sojourn time in ℤ+ for the Bernoulli random walk on ℤ." ESAIM: Probability and Statistics 16 (2012): 324-351. <http://eudml.org/doc/222482>.

@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; random walk},
language = {eng},
month = {8},
pages = {324-351},
publisher = {EDP Sciences},
title = {Sojourn time in ℤ+ for the Bernoulli random walk on ℤ},
url = {http://eudml.org/doc/222482},
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
DA - 2012/8//
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; random walk
UR - http://eudml.org/doc/222482
ER -

References

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