Necessary conditions for nonlinear functionals of Gaussian processes to satisfy central limit theorems

AbstractLet X = (Xt, t ϵ R) be a stationary Gaussian process on (Ω, F, P) with time-shift operators (Us, s ϵ R) and let H(X) = L2(Ω, σ(X), P) denote the space of square-integrable functionals of X. Say that Y ϵ H(X) with EY = 0 satisfies the Central Limit Theorem (CLT) if ZTʃ0T (USY) ds⧸Var ʃ0T (USY) ⧸ ds 12 D→N(0,1) as T→∞A family of martingales (ZT(t), t ⩾ 0) is exhibited for which ZT(∞) ≡ ZT, and martingale techniques and results are used to provide sufficient conditions on X and Y for the CLT. These conditions are then shown to be necessary for slightly more restrictive central limit behavior of Y