Prediction for discrete time series

Abstract

Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times $\lambda_n$ along which we will be able to estimate the conditional probability $P(X_{\lambda_n+1}=x|X_0,...,X_{\lambda_n})$ from data segment $(X_0,...,X_{\lambda_n})$ in a pointwise 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 (among others, all stationary and ergodic Markov chains are included in this class) then $ \lim_{n\to \infty} {n\over \lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $\lambda_n$ is upperbounded by a polynomial, eventually almost surely.