Forecasting for stationary binary time series

TL;DR

This paper introduces a method for predicting the next value in a stationary, ergodic binary time series at specific stopping times, ensuring consistency without prior distribution knowledge.

Contribution

The paper presents a novel prediction procedure that makes infinitely many predictions at carefully chosen times with proven consistency for stationary ergodic binary processes.

Findings

The procedure is consistent under certain conditions.

The growth rate of stopping times is estimated.

Predictions are made infinitely often at selected times.

Abstract

The forecasting problem for a stationary and ergodic 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. We show that the proposed procedure is consistent under certain conditions, and we estimate the growth rate of the stopping times.