Limitations on intermittent forecasting

TL;DR

This paper demonstrates fundamental limitations in pointwise intermittent forecasting for stationary and ergodic binary time series, showing that certain estimations cannot be universally achieved along stopping times.

Contribution

It proves that, unlike Markov chains, general stationary ergodic binary processes cannot be universally predicted at stopping times in a pointwise manner.

Findings

Forecasting is impossible for all stationary ergodic binary series.

Universal estimation along stopping times fails in the general case.

Markov chains are an exception where prediction is feasible.

Abstract

Bailey showed that the general pointwise forecasting for stationary and ergodic time series has a negative solution. However, it is known that for Markov chains the problem can be solved. Morvai showed that there is a stopping time sequence $\{\lambda_n\}$ such that $P(X_{\lambda_n+1}=1|X_0,...,X_{\lambda_n}) $ can be estimated from samples $(X_0,...,X_{\lambda_n})$ such that the difference between the conditional probability and the estimate vanishes along these stoppping times for all stationary and ergodic binary time series. We will show it is not possible to estimate the above conditional probability along a stopping time sequence for all stationary and ergodic binary time series in a pointwise sense such that if the time series turns out to be a Markov chain, the predictor will predict eventually for all $n$.