Nonparametric sequential prediction for stationary processes

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.Comment: Published in at http://dx.doi.org/10.1214/10-AOP576 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org