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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  4. K.-L. Chung and W. Feller, On fluctuations in coin-tossings. Proc. Natl. Acad. Sci. USA35 (1949) 605–608.  
  5. W. Feller, An introduction to probability theory and its applications I, 3rd edition. John Wiley & Sons (1968).  
  6. P. Flajolet and R. Sedgewick, Analytic combinatorics. Cambridge University Press, Cambridge (2009).  
  7. A. Lachal, arXiv:1003.5009[math.PR] (2010).  
  8. A. Rényi, Calcul des probabilités. Dunod (1966).  
  9. E. Sparre Andersen, On the number of positive sums of random variables. Skand. Aktuarietidskrift (1949) 27–36.  
  10. E. Sparre Andersen, On the fluctuations of sums of random variables I-II. Math. Scand. 1 (1953) 263–285; 2 (1954) 195–223.  
  11. F. Spitzer, Principles of random walk, 2nd edition. Graduate Texts in Mathematics34 (1976).  

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