Limits to consistent on-line forecasting for ergodic time series

TL;DR

This paper explores fundamental limitations in online forecasting of ergodic time series, demonstrating that many prediction problems are inherently unsolvable under minimal assumptions, contrasting with known results under stronger mixing conditions.

Contribution

It provides new negative results showing the impossibility of certain prediction tasks for ergodic time series, extending the understanding of fundamental limits in time-series forecasting.

Findings

Many plausible forecasting problems are unsolvable.

Predictors consistent under mixing conditions fail in ergodic settings.

Survey of related results and new derivations of impossibility theorems.

Abstract

This study concerns problems of time-series forecasting under the weakest of assumptions. Related results are surveyed and are points of departure for the developments here, some of which are new and others are new derivations of previous findings. The contributions in this study are all negative, showing that various plausible prediction problems are unsolvable, or in other cases, are not solvable by predictors which are known to be consistent when mixing conditions hold.