Prediction for discrete time series

TL;DR

This paper introduces a method to estimate conditional probabilities in stationary ergodic time series at specific stopping times, applicable to a broad class including Markovian processes, with convergence guarantees.

Contribution

It proposes a sequence of stopping times for pointwise consistent probability estimation in stationary ergodic processes, extending applicability to countably infinite alphabets and Markovian processes.

Findings

Consistent estimation at stopping times for a broad class of processes.

Almost sure convergence of the estimators.

Polynomial upper bound on stopping times for finite entropy processes.

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.