A note on prediction for discrete time series

Let {Xn} be a stationary and ergodic time series taking values from a finite or countably infinite set X and that f(X) is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times λn along which we will be able to estimate the conditional expectation E(f(Xλn+1)|X0, . . . , Xλn) from the observations (X0, . . . , Xλn) in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (in particular, all stationary and ergodic Markov chains are included in this class) then limn→∞ n λn > 0 almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λn is upper bounded by a polynomial, eventually almost surely.