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Shepp statistic for Markov chains application to a long-run average cost criterion

Published online by Cambridge University Press:  14 July 2016

Brahim Ksir
Brahim Ksir*
Affiliation:
University of Constantine
*
∗ Postal address: Institut de Mathématiques, Université de Constantine, 25000 Constantine, Algeria.
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Abstract

This paper is a generalization to Markov chains of the work of Shepp [6] in the i.i.d case. Shepp studies the limiting values of the averages Tn = (Sn + f(n) – Sn )/f(n) where Sn = X0 + X 1+ · ·· + Xn , X 0 = 0, n = 1, 2, ···, is a sum of mutually independent and identically distributed random variables. The function f takes positive integer values and non-decreasingly tends to infinity. Here we take a class of functions f in central position f(n) = [c log n], c > 0, n = 1, 2, ···. There are many refinements of the function f in the i.i.d case [1], [2]. Here we consider the more general case where X 1 , · ··, Xn is an irreducible and recurrent Markov chain. The state space of the chain is either compact or countable.

Keywords

RECURRENT MARKOV CHAINLONG-RUN AVERAGE COST CRITERION

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 4 , December 1989, pp. 767 - 775
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

Research partly carried out while the author was on leave at LSTA, Université de Paris VI.

References

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