# 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/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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- K.-L. Chung and W. Feller, On fluctuations in coin-tossings. Proc. Natl. Acad. Sci. USA35 (1949) 605–608.
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- F. Spitzer, Principles of random walk, 2nd edition. Graduate Texts in Mathematics34 (1976).

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