Strongly consistent Online forecasting of centered Gaussian processes
An estimator (E) over cap (dn, n) of the conditional expectation E[Xn+1\X-n,... Xn-dn+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 X-1,..., X.. For a suitable choice of the length d(n) --> infinity (n --> infinity) of the past covered by the conditional expectation, it is established that \(E) over cap (dn, n) - E[Xn+1\ X-n,..., Xn-dn+1]\ --> 0 with probability 1. In addition, sufficient conditions for \E[Xn+1\ X-n, Xn-1,...] - E[Xn+1\X-n,..., Xn-dn+1]\ --> 0 to hold with probability I are given, that is, conditions under which (E) over cap (dn,n) can be used as a strongly consistent forecaster for E[Xn+1 \ X-n, Xn-1,...].
gaussian process conditional expectation nonparametric forecasting of time series strong (universal) consistency