Nonparametric sequential prediction for stationary processes

We study the problem of finding an universal estimation scheme $h_n:{\mathbb{R}}^n\to \mathbb{R}$, $n=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$ 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 $1<p\le \mathrm{\infty }$, and show that for $p=1$ mere integrability does not suffice but $L{\log }^{+}L$ does.