Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T06:49:40.521Z Has data issue: false hasContentIssue false
  • English
  • Français

A bernoulli excursion and its various applications

Published online by Cambridge University Press:  01 July 2016

Lajos Takács
Lajos Takács*
Affiliation:
Case Western Reserve University
*
∗Postal address: 2410 Newbury Drive, Cleveland Heights, OH 44118, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with a random walk process in which and for i = 1, 2, ···, 2n. This process is called a Bernoulli excursion. The main object is to find the distribution, the moments, and the asymptotic distribution of the random variable ω n defined by . The results derived have various applications in the theory of probability, including random graphs, tournaments and order statistics.

Keywords

BROWNIAN EXCURSIONRANDOM GRAPHSORDER STATISTICSTOURNAMENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 557 - 585
Copyright
Copyright © Applied Probability Trust 1991 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
2. Biane, Ph. and Yor, M. (1987) Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. 111, 23–101.Google Scholar
3. Carleman, T. (1922) Sur le problème des moments. C. R. Acad. Sci. Paris 174, 1680–1682.Google Scholar
4. Cohen, J. W. and Hooghiemstra, G. (1981) Brownian excursion, the M/M/1 queue and their occupation times. Math. Operat. Res. 6, 608–629.CrossRefGoogle Scholar
5. Darling, D. A. (1983) On the supremum of a certain Gaussian process. Ann. Prob. 11, 803–806.Google Scholar
6. Debruijn, N. G. and Morselt, B. J. M. (1967) A note on plane trees. J. Comb. Theory 2, 27–34.Google Scholar
7. Frechet, M. and Shohat, J. (1931) A proof of the generalized second limit theorem in the theory of probability. Trans. Amer. Math. Soc. 33, 533–543.Google Scholar
8. Getoor, R. K. and Sharpe, M. J. (1979) Excursions of Brownian motion and Bessel processes. Z. Wahrscheinlichkeitsth. 47, 83–106.CrossRefGoogle Scholar
9. Gikhman, I. I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes . W. B. Saunders, Philadelphia.Google Scholar
10. Groeneboom, P. (1989) Brownian motion with a parabolic drift. Prob. Theory Rel. Fields 81, 79–109.CrossRefGoogle Scholar
11. Guy, R. K. (1988) The strong law of small numbers. Amer. Math. Monthly 95, 697–712.CrossRefGoogle Scholar
12. Harary, F., Prins, G. and Tutte, W. T. (1964) The number of plane trees. Nederl. Akad. Wetensch. Proc. Ser. A 67, 319–329.CrossRefGoogle Scholar
13. Hardy, G. H. and Wright, E. M. (1956) An Introduction to the Theory of Numbers , 3rd edn. Oxford University Press.Google Scholar
14. Holt, D. R. and Crow, E. L. (1973) Tables and graphs of the stable probability density functions. J. Res. Nat. Bureau Stand. B Math. Sci. 77B, 143–198.CrossRefGoogle Scholar
15. Klarner, D. A. (1969) A correspondence between two sets of trees. Nederl. Akad. Wetensch. Proc. Ser. A 72, 292–296.CrossRefGoogle Scholar
16. Klarner, D. A. (1970) Correspondences between plane trees and binary sequences. J. Comb. Theory 9, 401–411.CrossRefGoogle Scholar
17. Kleitman, D. (1970) The number of tournament score sequences for a large number of players. In Combinatorial Structures and their Applications , ed. Guy, R., Hanani, H., Sauer, N. and Schonheim, J.. Gordon and Breach, New York, pp. 209–213.Google Scholar
18. Louchard, G. (1984) Kac's formula, Lévy's local time and Brownian excursion. J. Appl. Prob. 21, 479–499.Google Scholar
19. Louchard, G. (1984) The Brownian excursion area: a numerical analysis. Comput. Math. Appl. 10, 413–417. (Erratum: Ibid. A 12 (1986), 375.) CrossRefGoogle Scholar
20. Miller, J. C. P. (1946) The Airy Integral. Cambridge University Press.Google Scholar
21. Moon, J. W. (1968) Topics on Tournaments. Holt, Rinehart, and Winston, New York.Google Scholar
22. Narayana, T. V. and Bent, D. H. (1964) Computation of the number of score sequences in round-robin tournaments. Canad. Math. Bull. 7, 133–136.Google Scholar
23. Odlyzko, A. M. and Wilf, H. S. (1988) The editor's corner: n coins in a fountain. Amer. Math. Monthly 95, 840–843.Google Scholar
24. Ramanujan, S. (1919) Proof of certain identities in combinatorial analysis. Proc. Camb. Phil. Soc. 19, 214–216. [Collected Papers of Srinivasa Ramanujan , ed. Hardy, G. H., Seshu Aiyar, P. V. and Wilson, B. M.. Cambridge University Press (1927), pp. 214–215. Reprinted (1962) by Chelsea, New York.] Google Scholar
25. Riordan, J. and Sloane, N. J. A. (1969) The enumeration of rooted trees by total height. J. Austral. Math. Soc. 10, 278–282.Google Scholar
26. Sarkadi, K. (1954) Mozdonyok várakozási idejérol. (On the waiting time of locomotives.) A Magyar Tudományos Akadémia Alkalmazott Matematikai Intézetének Közleményei 3, 191–194.Google Scholar
27. Slater, L. J. (1960) Confluent Hypergeometric Functions. Cambridge University Press.Google Scholar
28. Szekeres, G. (1968) A combinatorial interpretation of Ramanujan's continued fraction. Canad. Math. Bull. 11, 405–408.Google Scholar
29. Takács, L. (1956) Egy közlekedéssel kapcsolatos valószinuségszámitási problémáról. (On a probability problem connected with railway traffic.) Publ. Math. Inst. Hungar. Acad. Sci. 1, 99–107.Google Scholar
30. Takács, L. (1986) Some asymptotic formulas for lattice paths. J. Statist. Planning Inf. 14, 123–142.CrossRefGoogle Scholar
31. Vervaat, W. (1979) A relation between Brownian bridge and Brownian excursion. Ann. Prob. 7, 143–149.CrossRefGoogle Scholar
32. Voloshin, Ju. M. (1974) Enumeration of the terms of the object domains according to the depth of embedding. Soviet Math. Dokl. 15, 1777–1782.Google Scholar
33. Watson, G. S. (1961) Goodness-of-fit test on a circle. Biometrika 48, 109–114.Google Scholar
34. Whitworth, W. A. (1879) Arrangements of m things of one sort and n things of another sort, under certain conditions of priority. Messenger of Math. 8, 105–114.Google Scholar
35. Whitworth, W. A. (1886) Choice and Chance , 4th edn. Deighton Bell, Cambridge.Google Scholar
36. Winston, K. J. and Kleitman, D. J. (1983) On the asymptotic number of tournament score sequences. J. Comb. Theory A 35, 208–230.CrossRefGoogle Scholar
37. Wolfram, S. (1988) Mathematica. A System for Doing Mathematics by Computers. Addison-Wesley, Redwood City, CA.Google Scholar
38. Zolotarev, V. M. (1986) One-dimensional Stable Distributions. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar