Nonparametric sequential prediction for stationary processes

TL;DR

This paper develops a universal nonparametric prediction scheme for stationary ergodic processes that guarantees almost sure convergence of the average prediction error for all processes in certain integrability classes.

Contribution

It constructs a single prediction scheme applicable to all stationary ergodic processes in L^p, with specific conditions for p=1 involving L log^+ L.

Findings

Universal prediction scheme for all L^p stationary ergodic processes.

Convergence of average prediction error to zero almost surely.

Mere integrability is insufficient for p=1; L log^+ L is required.

Abstract

We study the problem of finding an universal estimation scheme $h_n:\mathbb{R}^n\to \mathbb{R}$, $n=1,2,...$ which will satisfy \lim_{t\rightarrow\infty}{\frac{1}{t}}\sum_{i=1}^t|h_ i(X_0,X_1,...,X_{i-1})-E(X_i|X_0,X_1,...,X_{i-1})|^p=0 a.s. for all real valued stationary and ergodic processes that are in $L^p$. We will construct a single such scheme for all $1<p\le\infty$, and show that for $p=1$ mere integrability does not suffice but $L\log^+L$ does.