Predicting the outcomes of every process for which an asymptotically accurate stationary predictor exists is impossible

TL;DR

The paper proves that it is impossible to create a universal predictor that accurately forecasts outcomes for all stationary ergodic processes with existing asymptotically accurate predictors, highlighting fundamental limitations in prediction theory.

Contribution

It establishes a negative result showing the non-existence of a universal predictor for all stationary ergodic sources with asymptotically accurate predictors.

Findings

Universal prediction for all such sources is impossible.

Contrasts with previous positive results for smaller process classes.

Highlights fundamental limitations in predictive modeling.

Abstract

The problem of prediction consists in forecasting the conditional distribution of the next outcome given the past. Assume that the source generating the data is such that there is a stationary ergodic predictor whose error converges to zero (in a certain sense). The question is whether there is a universal predictor for all such sources, that is, a predictor whose error goes to zero if any of the sources that have this property is chosen to generate the data. This question is answered in the negative, contrasting a number of previously established positive results concerning related but smaller sets of processes.