Summary
Let X=(X t ,t∈ℝ) be a stationary Gaussian process on (Ω, ℱ,P), letH(X) be the Hilbert space of variables inL 2 (Ω,P) which are measurable with respect toX, and let (U s ,s∈ℝ) be the associated family of time-shift operators. We sayY∈H(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if
in
[respectively,
], where
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).
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Chambers, D., Slud, E. Central limit theorems for nonlinear functionals of stationary Gaussian processes. Probab. Th. Rel. Fields 80, 323–346 (1989). https://doi.org/10.1007/BF01794427
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DOI: https://doi.org/10.1007/BF01794427