Nonparametric sequential prediction for stationary processes

Abstract

We study the problem of finding an universal estimation scheme h_n : ℝⁿ→ℝ, n=1, 2, … which will satisfy

lim_t→∞(1/t)∑_i=1^t|h_i(X₀, X₁, …, X_i−1)−E(X_i|X₀, X₁, …, 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≤∞, and show that for p=1 mere integrability does not suffice but L log⁺ L does.