Abstract
The solutions of various problems in the theories of queuing processes, branching processes, random graphs and others require the determination of the distribution of the sojourn time (occupation time) for the Brownian excursion. However, no standard method is available to solve this problem. In this paper we approximate the Brownian excursion by a suitably chosen random walk process and determine the moments of the sojourn time explicitly. By using a limiting approach, we obtain the corresponding moments for the Brownian excursion. The moments uniquely determine the distribution, enabling us to derive an explicit formula.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover: New York, 1970.
K. L. Chung, “Excursions in brownian motion,” Arkiv för Matematik vol. 14 pp. 157–179, 1976.
J. W. Cohen and G. Hooghiemstra, “Brownian excursion, the M/M/1 queue and their occupation times,” Mathematics of Operations Research vol. 6 pp. 608–629, 1981.
Faà di Bruno, “Note sur une nouvelle formule du calcul differential,” Quarterly Journal of Mathematics vol. 1 pp. 359–360, 1855.
M. D. Donsker, “An invariance principle for certain probability limit theorems,” Four Papers on Probability. Memoirs of the American Mathematical Society vol. 6 pp. 1–12, 1951.
I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, W.B. Saunders: Philadelphia, 1969.
Ch. Jordan, Calculus of Finite Differences, Budapest, 1939. [Reprinted by Chelsea: New York, 1947.]
P. S. Laplace, Théorie Analytique des Probabilités, Courcier, Paris, 1812. [Reprinted by Culture et Civilization: Bruxelles, 1962.]
L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.
L. Takács, “Conditional limit theorems for branching processes,” Journal of Applied Mathematics and Stochastic Analysis vol. 4 pp. 263–292, 1991.
L. Takács, “The asymptotic distribution of the total heights of random rooted trees,” Acta Scientiarum Mathematicarum (Szeged) vol. 57 pp. 613–625, 1993.
L. Takács, “Brownian local times,” Journal of Applied Mathematics and Stochastic Analysis vol. 8 pp. 209–232, 1995a.
L. Takács, “On the local time of the Brownian motion,” The Annals of Applied Probability vol. 5 pp. 741–756, 1995b.
L. Takács, “Queueing methods in the theory of random graphs,” Advances in Queueing Theory, Methods, and Open Problems, Edited by J. H. Dshalalow. CRC Press: Boca Raton, FL, pp. 45–78, 1995c.
I. Todhunter, “A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace,” Cambridge University Press, 1865. [Reprinted by Chelsea: New York, 1949.]
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Taka´cs, L. The Distribution of the Sojourn Time for the Brownian Excursion. Methodology and Computing in Applied Probability 1, 7–28 (1999). https://doi.org/10.1023/A:1010060107265
Issue Date:
DOI: https://doi.org/10.1023/A:1010060107265