Abstract
This paper is concerned with two stochastic processes; namely, a Bernoulli excursion and a tied-down random walk. For each process a random variable is defined as the area of a random set bounded by the real line and the sample function of the process. The results derived for random walks are applied to the theory of random trees to determine the distribution and the asymptotic distribution of the total height of a tree, and to the theory of order statistics to determine the asymptotic behavior of the moments and the distribution of a statistic which measures the deviation between two empirical distribution functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramowitz M. and Stegun I.A., Handbook of Mathematical Functions, Dover, New York, 1965.
Biane Ph. and Yor M., Valeurs principales associées aux temps locaux browniens, Bulletin des Sciences Mathématiques, 111 (1987), 23–101.
Carleman T., Sur le problème des moments, Comptes Rendus Acad. Sci. Paris, 174 (1922), 1680–1682.
Carleman T., Les Fonctions Quasi-Analytiques, Gauthier-Villars, Paris, 1926.
Cohen J.W. and Hooghiemstra G., Brownian excursions, the M/M/1 queue and their occupation times, Mathematics of Operations Research, 6 (1981), 608–629.
Darling D.A., On the supremum of a certain Gaussian process, The Annals of Probability, 11 (1983), 803–806
Fréchet M. and Shohat J., A proof of the generalized second-limit theorem in the theory of probability, Transactions of the American Mathematical Society, 33 (1931), 533–543.
Getoor R.K. and Sharpe M.J., Excursions of Brownian motion and Bessel processes, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 47 (1979), 83–106.
Groeneboom P., Brownian motion with parabolic drift, Probability Theory and Related Fields, 81 (1989), 79–109.
Hardy G.H. and Wright E.M., An Introduction to the Theory of Numbers, Third edition, Oxford University Press, 1956.
Johnson B.McK. and Killeen T., An explicit formula for the c.d.f. of the L 1 norm of the Brownian bridge, The Annals of Probability, 11 (1983), 807–808.
Louchard G., Kac’s formula, Lévy’s local time and Brownian excursion, Journal of Applied Probability, 21 (1984), 479–499.
Louchard G., The Brownian excursion area: a numerical analysis, Computers and Mathematics with Applications 10 (1984), 413–417. [Erratum: Ibid A 12 (1986), 375].
Miller J.C.P., The Airy Integral, Cambridge University Press, 1946.
Ramanujan S., Proof of certain identities in combinatorial analysis, Proceedings of the Cambridge Philosophical Society, 19 (1919), 214–216.
Rice S.O., The integral of the absolute value of the pinned Wiener process—Calculation of its probability density by numerical integration, The Annals of Probability, 10 (1982), 240–243.
Shepp L.A., On the integral of the absolute value of the pinned Wiener process, The Annals of Probability, 10 (1982), 234–239.
Slater L.J., Confluent Hypergeometric Functions, Cambridge University Press, 1950.
Szekeres G., A combinatorial interpretation of Ramanujan’s continued fraction, Canadian Mathematical Bulletin, 11 (1968), 405–408.
Voloshin Ju.M., Enumeration of the terms of the object domains according to the depth of embedding, Soviet Mathematics-Doklady, 15 (1974), 1777–1782.
Wolfram S., Mathematica. A System for Doing Mathematics by Computer, Addison-Wesley, Redwood City, California, 1988.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Takács, L. (1992). Random Walk Processes and their Various Applications. In: Galambos, J., Kátai, I. (eds) Probability Theory and Applications. Mathematics and Its Applications, vol 80. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2817-9_1
Download citation
DOI: https://doi.org/10.1007/978-94-011-2817-9_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5252-8
Online ISBN: 978-94-011-2817-9
eBook Packages: Springer Book Archive