Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T05:44:57.730Z Has data issue: false hasContentIssue false
  • English
  • Français

On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

M. S. Sgibnev
M. S. Sgibnev*
Affiliation:
Russian Academy of Sciences
*
∗Postal address: Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 90, 630090 Russia. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with submultiplicative moments for the stationary distributions π of some Markov chains taking values in ℝ+ or ℝ which are closely related to the random walks generated by sequences of independent identically distributed random variables. Necessary and sufficient conditions are given for ∫φ(x)π(dx) < ∞, where φ(x) is a submultiplicative function, i.e. φ(0) = 1 and φ(x+y) ≤ φ(x)φ(y) for all x, y.

Keywords

Oscillating random walkgeneralized Lindley processstationary distributionrandom walksupremumsubmultiplicative functionsmoments

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 78 - 85
Copyright
Copyright © Applied Probability Trust 1999 

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.)

Footnotes

This research was supported by Grant 96-01-01939 of the Russian Foundation for Fundamental Research.

References

Alsmeyer, G. (1992). On generalized renewal measures and certain first passage times. Ann. Prob. 20, 1229–1247.Google Scholar
Borovkov, A. A. (1980). A limit distribution for an oscillating random walk. Theory Prob. Appl. 25, 649–651.Google Scholar
Borovkov, A. A., and Korshunov, D. A. (1994). Ergodicity in a sense of weak convergence, equilibrium-type identities and large deviations for Markov chains. In Probability Theory and Mathematical Statistics, Proc. 6th Vilnius Conference, ISP BV/TEV Ltd, Utrecht, pp. 89–98.Google Scholar
Borovkov, A. A., and Korshunov, D. A. (1996). Probabilities of large deviations for one-dimensional Markov chains. I. Stationary distributions. Theory Prob. Appl. 41, 1–24.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications II. Wiley, New York.Google Scholar
Hille, E., and Phillips, R. S. (1957). Functional Analysis and Semi-Groups (AMS Colloquium Publications 31). AMS, Providence, RI.Google Scholar
Kemperman, J. H. B. (1974). The oscillating random walk. Stoch. Process. Appl. 2, 1–29.Google Scholar
Kiefer, J., and Wolfowitz, J. (1956). On the characteristics of the general queueing process, with applications to random walk. Ann. Math. Statist. 27, 147–161.CrossRefGoogle Scholar
Sgibnev, M. S. (1997). Submultiplicative moments of the supremum of a random walk with negative drift. Statist. Prob. Lett. 32, 377–384.Google Scholar
Tweedie, R. L. (1983). The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.CrossRefGoogle Scholar