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

top
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

top

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

top
  1. [1] P. Billingsley, Convergence of probability measures. John Wiley & Sons (1968). Zbl0944.60003MR233396
  2. [2] A.-N. Borodin and P. Salminen, Handbook of Brownian motion – facts and formulae, Probability and its Applications. Birkhäuser Verlag (1996). Zbl1012.60003MR1477407
  3. [3] V. Cammarota, A. Lachal and E. Orsingher, Some Darling-Siegert relationships connected with random flights. Stat. Probab. Lett.79 (2009) 243–254. Zbl1181.33005MR2483547
  4. [4] K.-L. Chung and W. Feller, On fluctuations in coin-tossings. Proc. Natl. Acad. Sci. USA35 (1949) 605–608. Zbl0037.36310MR33459
  5. [5] W. Feller, An introduction to probability theory and its applications I, 3rd edition. John Wiley & Sons (1968). Zbl0039.13201MR228020
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.