Probability Theory and Related Fields

, Volume 80, Issue 3, pp 323–346

# Central limit theorems for nonlinear functionals of stationary Gaussian processes

• Daniel Chambers
• Eric Slud
Article

DOI: 10.1007/BF01794427

Chambers, D. & Slud, E. Probab. Th. Rel. Fields (1989) 80: 323. doi:10.1007/BF01794427

## Summary

Let X=(Xt,t∈ℝ) be a stationary Gaussian process on (Ω, ℱ,P), letH(X) be the Hilbert space of variables inL2 (Ω,P) which are measurable with respect toX, and let (Us,s∈ℝ) be the associated family of time-shift operators. We sayYH(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively,], where
$$Y_T (t) \equiv {{\int\limits_0^{Tt} {U_s \circ Yds} } \mathord{\left/ {\vphantom {{\int\limits_0^{Tt} {U_s \circ Yds} } {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right. \kern-\nulldelimiterspace} {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$
andW(•) is a standard Wiener process on [0,1]. This paper provides some general sufficient conditions onX andY ensuring thatY satisfies the CLT or FCLT. Examples ofY are given which satisfy the CLT but not the FCLT. This work extends CLT's of Maruyama (1976) and Breuer and Major (1983).