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 + X1+ · ·· + Xn, X0 = 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 X1, · ··, Xn is an irreducible and recurrent Markov chain. The state space of the chain is either compact or countable.
