Central limit theorems for nonlinear functionals of stationary Gaussian processes

Abstract

Summary. Let X = (~, t ~IR) be a stationary Gaussian process on (f2, Y, P), let H(X) be the Hilbert space of variables in L 2 (s P) which are measurable with respect to X, and let (Us, sslR) be the associated family of time-shift operators. We say Yell(X) (with E(Y)= 0) satisfies the functional central Maruyama (1976) and Breuer and Major (1983).

Multiple Wiener-itô integral expansions for level-crossing-count functionals

“…After expressing the functional Itx o > o] as a multiple Wiener-It6 integral expansion and expressing products of expansions through the Diagram Theorem, we obtain the first Proposition and Corollary. C=+l,(mZj~Jdp ~(l_p2)m_j+l/2. In the following theorem, asymptotic normality has been proved by Ho and Sun (1987) and also follows easily from Theorems 1 and 2 of Chambers and Slud (1989a) using the representation of Proposition 1. The positive lower bound for the asymptotic variance is apparently new.…”

“…Just as we obtained Theorem 1 from Proposition 1 by means of the general central limit theorem of Chambers and Slud (1989a), so from Theorem 2 follows the central limit theorem of Cuzick (1976) …”

“…13 of Sinai 1977, for clarification) that spectral ergodic-theoretic properties such as weak-mixing and mixing for a functional ("factor" in the language of ergodic theory) of a stationary Gaussian process, can be expressed in terms of the absolute-continuity equivalence class of the underlying spectral measure together with the multiple Wiener-It6 integrands. A second way to exploit Wiener-It5 integral expansions has been developed by Taqqu (1975), Dobrushin and Major (1979), Maruyama (1976) and Chambers and Slud (1989a, b) among others 9 These authors prove general (functional) central and noncentral limit theorems for such expansions 9 In the present paper, central limit theorems of Chambers and Slud (1989a) will be applied to the level-crossings counts.…”

Critical points of multidimensional random Fourier series: Variance estimates

“…We want to mention here the pioneering contributions of Chambers and Slud [14], Slud [38,39], Kratz and León [24], Sodin and Tsirelson [40].…”

Level crossings and other level functionals of stationary Gaussian processes

“…• Slud (1991,1994) Introducing the theoretical tool of Multiple Wiener Itô integrals (MWI's) allowed some authors as Taqqu (see [154]), Dobrushin and Major (see [50]), Giraitis and Surgailis (see [56]), Maruyama (see [103]), Chambers and Slud (see [34]), ... to prove general functional central limit theorems (FCLT) (and non central limit theorems) for MWI expansions. In the 1990s, Slud (see [147]) applied a general central limit theorem of Chambers and Slud (see [34]) to provide Cuzick's CLT for the zero crossings N t (0) by the process X, without needing Cuzick's additional assumptions to get a strictly positive limiting variance.…”