Sojourn time in ℤ + for the Bernoulli random walk on ℤ
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 324-351

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.

DOI : 10.1051/ps/2010013
Classification : 60G50, 60J22, 60J10, 60E10
Keywords: random walk, sojourn time, generating function 
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     author = {Lachal, Aim\'e},
     title = {Sojourn time in $\mathbb {Z}^{+}$ for the {Bernoulli} random walk on $\mathbb {Z}$},
     journal = {ESAIM: Probability and Statistics},
     pages = {324--351},
     year = {2012},
     publisher = {EDP Sciences},
     volume = {16},
     doi = {10.1051/ps/2010013},
     zbl = {1275.60046},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2010013/}
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Lachal, Aimé. Sojourn time in $\mathbb {Z}^{+}$ for the Bernoulli random walk on $\mathbb {Z}$. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 324-351. doi: 10.1051/ps/2010013

[1] P. Billingsley, Convergence of probability measures. John Wiley & Sons (1968). | Zbl | MR

[2] A.-N. Borodin and P. Salminen, Handbook of Brownian motion - facts and formulae, Probability and its Applications. Birkhäuser Verlag (1996). | Zbl | MR

[3] V. Cammarota, A. Lachal and E. Orsingher, Some Darling-Siegert relationships connected with random flights. Stat. Probab. Lett. 79 (2009) 243-254. | Zbl | MR

[4] K.-L. Chung and W. Feller, On fluctuations in coin-tossings. Proc. Natl. Acad. Sci. USA 35 (1949) 605-608. | Zbl | MR

[5] W. Feller, An introduction to probability theory and its applications I, 3rd edition. John Wiley & Sons (1968). | Zbl | MR

[6] P. Flajolet and R. Sedgewick, Analytic combinatorics. Cambridge University Press, Cambridge (2009). | Zbl | MR

[7] A. Lachal, arXiv:1003.5009[math.PR] (2010).

[8] A. Rényi, Calcul des probabilités. Dunod (1966). | Zbl

[9] E. Sparre Andersen, On the number of positive sums of random variables. Skand. Aktuarietidskrift (1949) 27-36. | Zbl | MR

[10] E. Sparre Andersen, On the fluctuations of sums of random variables I-II. Math. Scand. 1 (1953) 263-285; 2 (1954) 195-223. | Zbl | MR | EuDML

[11] F. Spitzer, Principles of random walk, 2nd edition. Graduate Texts in Mathematics 34 (1976). | Zbl | MR

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