Central limit theorems for nonlinear functionals of stationary Gaussian processes

Abstract

Let X=(Xt, t∈ℝ) be a stationary Gaussian process on (Ω, ℱ, P), let H(X) be the Hilbert space of variables in L2 (Ω, P) which are measurable with respect to X, and let (Us, s∈ℝ) be the associated family of time-shift operators. We say Y∈H(X) (with E(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if[Figure not available: see fulltext.] in[Figure not available: see fulltext.] [respectively,[Figure not available: see fulltext.]], where {Mathematical expression} and W(•) is a standard Wiener process on [0,1]. This paper provides some general sufficient conditions on X and Y ensuring that Y satisfies the CLT or FCLT. Examples of Y are given which satisfy the CLT but not the FCLT. This work extends CLT's of Maruyama (1976) and Breuer and Major (1983). © 1989 Springer-Verlag.