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/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 -

References

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  6. [6] P. Flajolet and R. Sedgewick, Analytic combinatorics. Cambridge University Press, Cambridge (2009). Zbl1165.05001MR2483235
  7. [7] A. Lachal, arXiv:1003.5009[math.PR] (2010). 
  8. [8] A. Rényi, Calcul des probabilités. Dunod (1966). Zbl0141.14702
  9. [9] E. Sparre Andersen, On the number of positive sums of random variables. Skand. Aktuarietidskrift (1949) 27–36. Zbl0041.45006MR32115
  10. [10] E. Sparre Andersen, On the fluctuations of sums of random variables I-II. Math. Scand. 1 (1953) 263–285; 2 (1954) 195–223. Zbl0058.12102MR58893
  11. [11] F. Spitzer, Principles of random walk, 2nd edition. Graduate Texts in Mathematics 34 (1976). Zbl0359.60003MR388547

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