Guessing the output of a stationary binary time series

Abstract: The forward prediction problem for a binary time series ${X_n}{n=0}^{\infty}$ is to estimate the probability that $X{n+1}=1$ based on the observations $X_i$, $0\le i\le n$ without prior knowledge of the distribution of the process ${X_n}$. 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 exhibited.