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Some joint distributions in Bernoulli excursions

Part of: Markov processes Combinatorial probability Limit theorems

Published online by Cambridge University Press:  14 July 2016

Endre Csáki
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Abstract

Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.

Keywords

RANDOM WALKEXCURSIONADDITIVE FUNCTIONAL

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60J55: Local time and additive functionals 60C05: Combinatorial probability
Type
Part 4 Random Walks
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 239 - 250
Copyright
Copyright © Applied Probability Trust 1994 

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References

Carlitz, L. (1972) Sequences, paths, ballot numbers. Fibonacci Quart. 10, 531–549.Google Scholar
Csáki, E. and Mohanty, S. G. (1986) Some joint distributions for conditional random walks. Canad. J. Statist. 14, 19–28.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications , Vol. II. Wiley, New York.Google Scholar
Gessel, I. (1980) A noncommutative generalization and q-analog of the Lagrange inversion formula. Trans. Amer. Math. Soc. 257, 455–482.Google Scholar
Handa, B. R. and Mohanty, S. G. (1980) On q-binomial coefficients and some statistical applications. SIAM J. Math. Anal. 11, 1027–1035.Google Scholar
Kaigh, W. D. (1978) An elementary derivation of the distribution of the maxima of Brownian meander and Brownian excursions. Rocky Mountain J. Math. 8, 641–645.Google Scholar
Kesten, H. (1962) Occupation times for Markov and semi-Markov chains. Trans. Amer. Math. Soc. 103, 82–112.Google Scholar
Korolyuk, V. S. and Borovskikh, Yu. V. (1981) Asymptotic approximation of distribution of Wilcoxon statistics. Izv. Akad. Nauk. UzSSR Ser. Fiz.-Mat. Nauk , No. 2, 26–31, 98 (in Russian).Google Scholar
Krattenthaler, C. (1989a) Counting lattice paths with a linear boundary I. Sitz. Ber. d. öaw, Math./Naturwiss. Kl. 198, 87–107.Google Scholar
Krattenthaler, C. (1989b) Counting lattice paths with a linear boundary II: q-ballot and q-Catalan numbers. Sitz. Ber. d. öaw, Math./Naturwiss. Kl. 198, 171–199.Google Scholar
Krattenthaler, C. and Mohanty, S. G. (1993) On lattice path counting by major index and descents. Europ. J. Combinatorics 14, 43–51.Google Scholar
Pitman, J. and Yor, M. (1992) Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. 65, 326–356.Google Scholar
Revesz, P. (1990) Random Walk in Random and Non-Random Environments. World Scientific, Singapore.Google Scholar
Sinai, Ya. G. (1992) Distribution of some functionals of the integral of random walk. Teoret. i Mat. Fiz. 90, 323–353 (in Russian with English summary).Google Scholar
Szekely, G. J. (1977) The Brownian taboo bridge with an application in the theory of nonparametric statistics. In Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians (Tech. Univ. Prague, 1974) . Academia, Prague, Vol. A, 563–566.Google Scholar
Takács, L. (1986) Some asymptotic formulas for lattice paths. J. Statist. Planning Inf. 14, 123–142.Google Scholar
Takács, L. (1991a) A Bernoulli excursion and its various applications. Adv. Appl. Prob. 23, 557–585.Google Scholar
Takács, L. (1991b) On the distribution of the number of vertices in layers of random trees. J. Appl. Math. Stoch. Anal. 4, 175–186.Google Scholar
Takács, L. (1991C) Conditional limit theorems for branching processes. J. Appl. Math. Stoch. Anal. 4, 263–292.Google Scholar
Takács, L. (1992a) Random walk processes and their applications in order statistics. Ann. Appl. Prob. 2, 435–459.Google Scholar
Takács, L. (1992b) On the total heights of random rooted trees. J. Appl. Prob. 29, 543–556.Google Scholar
Winston, K. J. and Kleitman, D. J. (1983) On the asymptotic number of tournament score sequences. J. Combinat. Theory A35, 208–230.Google Scholar