Strongly consistent online forecasting of centered Gaussian processes

An estimator Eˆ(d_n,n) of the conditional expectation E[X_n+1|X_n,...,X(n-d_n+1)] in a centered, stationary, and ergodic Gaussian process {X_i}_i with absolutely summable Wold coefficients is constructed on the basis of having observed X₁,...,X_n . For a suitable choice of the length dn→∞ (n→∞) of the past covered by the conditional expectation, it is established that |Eˆ(d_n,n)-E[X_n+1|X_n ,...,X(n-d_n+1)]|→0 with probability 1. In addition, sufficient conditions for |E[X_n+1|X_n,X _n-1,...]-E[X_n+1|X_n,...,X(n-d_n +1)]| →0 to hold with probability 1 are given, that is, conditions under which Eˆ(d_n,n) can be used as a strongly consistent forecaster for |E[X_n+1|X_n,X _n-1,...]