Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T05:21:59.454Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for sums of chain-dependent processes

Published online by Cambridge University Press:  14 July 2016

G. L. O'Brien
G. L. O'Brien*
Affiliation:
York University, Downsview, Ontario
Get access Rights & Permissions [Opens in a new window]

Abstract

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

Keywords

MARKOV CHAINSTATIONARY SEQUENCESTRONG LIMIT LAWSCENTRAL LIMIT THEOREM
Type
Short Communications
Information
Journal of Applied Probability , Volume 11 , Issue 3 , September 1974 , pp. 582 - 587
Copyright
Copyright © Applied Probability Trust 1974 

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] Davydov, Ju. A. (1969) On the strong mixing property for Markov chains with a countable number of states. Soviet Math. Dokl. 10, 825–827.Google Scholar
[2] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[3] Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1966) Denumerable Markov Chains. Van Nostrand, Princeton, New Jersey.Google Scholar
[4] Ibragimov, I. A. (1962) Some limit theorems for stationary processes. Theor. Probability Appl. 7, 349–382.CrossRefGoogle Scholar
[5] Janssen, J. (1969) Les processus (J-X) . Cahiers Centre Etudes Recherche Oper. 11, 181–214.Google Scholar
[6] O'Brien, G. L. and Denzel, G. E. (1974) Limit theorems for extreme values of chain-dependent processes. To appear.Google Scholar
[7] Resnick, S. I. and Neuts, M. F. (1970) Limit laws for maxima of a sequence of random variables defined on a Markov chain. Adv. Appl. Prob. 2, 323–343.CrossRefGoogle Scholar
[8] Reznik, M. H. (1968) The law of the iterated logarithm for some classes of stationary processes. Theor. Probability Appl. 13, 606–621.Google Scholar
[9] Rosenblatt, M. (1956) A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U.S.A. 42, 43–47.CrossRefGoogle Scholar
[10] Taga, Y. (1963) On limiting distributions of Markov renewal processes with finitely many states. Ann. Inst. Statist. Math. 18, 1–10.CrossRefGoogle Scholar