Necessary conditions for nonlinear functionals of Gaussian processes to satisfy central limit theorems (Q1263157)

Let \(X=\{X_ t:\) \(t\in {\mathbb{R}}\}\) be a stationary, centered Gaussian process on (\(\Omega\),\({\mathcal F},P)\), \(H(X)=L^ 2(\Omega,\sigma (X),P)\) the space of square-integrable functionals of X and \(\{U_ s:\) \(s\in {\mathbb{R}}\}\) the time-shift operators on H(X). Say that \(Y\in H(X)\) with \(E(Y)=0\) satisfies the CLT if \[ Z_ T:=(V_ Y(T))^{- 1/2}\int^{T}_{0}U_ s(Y)ds\to N(0,1)\quad in\quad distribution\quad (T\to \infty), \] where \(V_ Y(T)=E(\int^{T}_{0}U_ s(Y)ds)^ 2.\) Sufficient conditions for the validity of the CLT for Y are given which are also shown to be necessary for a slightly more restrictive limit behaviour. The paper shares with previous work by the authors [Probab. Theory Relat. Fields 80, 323-346 (1989)] and by \textit{G. Maruyama} [Proc. 3rd Japan-USSR Symp. Probab. Theory, Taschkent 1975, Lect. Notes Math. 550, 375-378 (1976; Zbl 0375.60045); Osaka J. Math. 22, 697-732 (1985; Zbl 0575.60025 ); Hokkaido Math. J. 15, 405-451 (1986; Zbl 0661.60036)] that Y is allowed to depend on infinitely many \(X_ t\) and that the conditions are expressed in terms of the representation of functionals in H(X) as sums of multiple Wiener-Itô integrals. A new feature is the use of results from martingale functional central limit theory through the introduction of suitable martingales \(\{Z_ T(t):t\geq 0\}\) such that \(Z_ T(\infty)=Z_ T\) and which have a stochastic integral representation related to the Wiener-Itô expansion of Y.