Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T02:26:36.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Simple random walk statistics. Part I: Discrete time results

Part of: Combinatorial probability 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 Wien, Austria.
∗Postal address: Department of Statistics, University of Economics, Augasse 2–6, A-1090 Wien, Austria.
Get access Rights & Permissions [Opens in a new window]

Abstract

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.

Keywords

LATTICE PATHSRANK ORDER STATISTICSDWASS'S METHOD

MSC classification

Secondary: 62G30: Order statistics; empirical distribution functions 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 311 - 330
Copyright
Copyright © Applied Probability Trust 1996 

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

Aneja, K. G. and Sen, Kanwar (1972) Random walk and distributions of rank order statistics. SIAM J. Appl. Math. 23, 276–287.Google Scholar
Barton, D. E. and Mallows, C. L. (1965) Some aspects of the random sequence. Ann. Math. Statist. 36, 236–260.Google Scholar
Böhm, W. and Panny, W. (1996) Simple random walk statistics. Part II: Continuous time results. J. Appl. Prob. 33, 331–339.Google Scholar
Chung, K. L. and Feller, W. (1949) Fluctuations in coin tossing. Proc. Natn. Acad. Sci. USA 35, 605–608.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Csáki, E. and Vincze, I. (1961) On some problems connected with the Galton test. Publ. Math. Inst. Hungarian Acad. Sci. Ser. A 6, 97–109.Google Scholar
Dwass, M. (1967) Simple random walk and rank order statistics. Ann. Math. Statist. 38, 1042–1053.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications. 3rd. edn. Wiley, New York.Google Scholar
Gnedenko, B. V. and Korolyuk, V. S. (1951) On the maximum discrepancy between two empirical distributions (in Russian). Dokl. Akad. Nauk SSSR. 80, 525–528.Google Scholar
Gnedenko, B. V. and Rvaceva, E. L. (1952) On a problem of the comparison of two empirical distributions (in Russian). Dokl. Akad. Nauk SSSR. 82, 513–516.Google Scholar
Sen, Kanwar (1965) On some combinatorial relations concerning the symmetric random walk. Publ. Math. Inst. Hungarian Acad. Sci. 9, 335–357.Google Scholar
Katzenbeisser, W. and Panny, W. (1984) Asymptotic results on the maximal deviation of simple random walks. Stoch. Proc. Appl. 18, 263–275.Google Scholar
Mihalevic, V. S. (1952) On the mutual disposition of two empirical distribution functions (in Russian). Dokl. Akad. Nauk SSSR. 85, 485–488.Google Scholar
Mohanty, S. G. (1979) Lattice Path Counting and Applications. Academic Press, New York.Google Scholar
Panny, W. (1984) The Maximal Deviation of Lattice Paths. Athenäum/Hain/Hanstein, Königstein.Google Scholar
ŠIdak, Z. (1973) Applications of random walks in non-parametric statistics. Bull. Inst. Int. Statist., Proc. 39. Session 45, Book 3, 34-43.Google Scholar
Vincze, I. (1957) Einige zweidimensionale Verteilungs- und Grenzverteilungssätze in der Theorie der geordneten Stichproben. Publ. Math. Inst. Hungarian Acad. Sci. 2, 183–203.Google Scholar