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1 results

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

  • Brahim Ksir
  • Journal:
    Journal of Applied Probability / Volume 26 / Issue 4 / December 1989
    Published online by Cambridge University Press:
    14 July 2016, pp. 767-775
    Print publication:
    December 1989

  • 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.