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

DOI: 10.1007/BF01794427

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


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
], 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).

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Daniel Chambers
    • 1
  • Eric Slud
    • 2
  1. 1.Department of MathematicsBoston CollegeChestnut HillUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA