Asymptotics for Prediction Errors of Stationaryprocesses with Reflection Positivity

Let H be the closed real linear hull of {Xs : s ∈ R} in L2(Ω,F , P ). Then H is a real Hilbert space with inner product (Y1, Y2) := E[Y1Y2] and norm ‖Y ‖ := (Y, Y ). For I ⊂ R, put HI for the closed subspace of H spanned by {Xs : s ∈ I}, and PI for the orthogonal projection operator of H onto HI . We write P⊥ I for the projection operator onto the orthogonal complement H ⊥ I of HI , i.e., P⊥ I Y = Y − PIY for Y ∈ H . Then, for T > 0 and t > 0, the projection P[−t,0]XT may be regarded as the best linear predictor of XT on the observations {Xs : −t ≤ s ≤ 0}, hence P⊥ [−t,0]XT = XT − P[−t,0]XT as its prediction error. Similarly, P⊥ (−∞,0]XT may be regarded as the prediction error for the prediction of XT on the observations {Xs : −∞ < s ≤ 0}.