Guessing the output of a stationary binary time series

The forward prediction problem for a binary time series {Xn} n=0 ∞ is to estimate the probability that Xn+1= 1 based on the observations X i , 0 ≤ i≤ n without prior knowledge of the distribution of the process {Xn}. It is known that this is not possible if one estimates at all values of n. We present a simple procedure which will attempt to make such a prediction infinitely often at carefully selected stopping times chosen by the algorithm. The growth rate of the stopping times is also studied.