Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T14:03:51.136Z Has data issue: false hasContentIssue false
  • English
  • Français

On the number of times where a simple random walk reaches its maximum

Part of: Approximations and expansions Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

W. Katzenbeisser  and
W. Panny
W. Katzenbeisser*
Affiliation:
University of Economics, Vienna
W. Panny*
Affiliation:
University of Economics, Vienna
*
∗Postal address: Department of Statistics, University of Economics, Augasse 2-6, A-1090 Vienna, Austria.
∗∗Postal address: Department of Computer Science, University of Economics, Augasse 2-6, A-1090 Vienna, Austria.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Qn denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn, are considered and their asymptotic behavior is studied.

Keywords

LATTICE PATHSRANK-ORDER STATISTICSKOLMOGOROV–SMIRNOV TESTASYMPTOTIC EXPANSIONS

MSC classification

Secondary: 62G30: Order statistics; empirical distribution functions 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 305 - 312
Copyright
Copyright © Applied Probability Trust 1992 

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

Dwass, M. (1967) Simple random walk and rank order statistics. Ann. Math. Statist. 38, 1042–1053.Google Scholar
Graham, R. L., Knuth, D. E. and Patashnik, O. (1989) Concrete Mathematics. Addison-Wesley, Reading, Mass.Google Scholar
Katzenbeisser, W. and Panny, W. (1984a) The moments of the test statistics Dn,n and D of the Kolmogorov-Smirnov test and their asymptotic behavior. In Proc. 4th Pannonian Symp. Math. Statist., ed. Grossmann, W., Pflug, G., Vince, I. and Wertz, E., 167–183, Akadémiai Kiadó, Budapest.Google Scholar
Katzenbeisser, W. and Panny, W. (1984b) Asymptotic results on the maximal deviation of simple random walks. Stoch. Proc. Appl. 18, 263–275.Google Scholar
Katzenbeisser, W. and Panny, W. (1990) Some further results on the height of lattice paths. J. Statist. Planning Inf. To appear.Google Scholar
Kemperman, J. H. B. (1959) Asymptotic expansions for the Smirnov test and for the range of cumulative sums. Ann. Math. Statist. 30, 448–462.CrossRefGoogle Scholar
Panny, W. (1984) The Maximal Deviation of Lattice Paths. Athenäum, Hain.Google Scholar
Panny, W. and Prodinger, H. (1985) The expected height of paths for several notions of height. Stud. Sci. Math. Hung. 20, 119–132.Google Scholar
Wilf, H. S. (1990) generatingfunctionology. Academic Press, Boston.Google Scholar