This paper is a generalization to Markov chains of the work of Shepp in the i.i.d. case. Shepp studies the limiting values of the averages Tn=(SnÅ+fÅ(nÅ)-Sn)/f(n) where Sn=X0+X1+ëëë+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)=[clogn], c>0, n=1,2,.... There are many refinements of the function f in the i.i.d. case. 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.
