Intermittent estimation of stationary time series

TL;DR

This paper introduces a simple, stopping-time-based algorithm for strongly consistent prediction of stationary time series, overcoming previous impossibility results by estimating only at selected times.

Contribution

It proposes a novel prediction method that makes predictions infinitely often at carefully chosen times, achieving strong consistency under certain conditions.

Findings

The algorithm is strongly consistent at selected stopping times.

It achieves $L_2$ consistency without additional conditions.

An upper bound on the growth rate of stopping times is provided.

Abstract

Let $\{X_n\}_{n=0}^{\infty}$ be a stationary real-valued time series with unknown distribution. Our goal is to estimate the conditional expectation of $X_{n+1}$ based on the observations $X_i$, $0\le i\le n$ in a strongly consistent way. Bailey and Ryabko proved that this is not possible even for ergodic binary time series if one estimates at all values of $n$. We propose a very simple algorithm which will make prediction infinitely often at carefully selected stopping times chosen by our rule. We show that under certain conditions our procedure is strongly (pointwise) consistent, and $L_2$ consistent without any condition. An upper bound on the growth of the stopping times is also presented in this paper.