Theorem (Prediction Error Upper Bound)

For the $m$-th order optimal prediction error (i.e. $E[e_{n}^{2}]$) we always have $$E[e_{n}^{2}]<E[X_{n}^{2}]$$unless $X_{n}$ and $X_{n-j}$ are uncorrelated (i.e. $E[X_{n}X_{n-j}]=0$) $\forall j=1,\dots,m$.