Central limit theorems for nonlinear functionals of stationary Gaussian processes
Central limit theorems for nonlinear functionals of stationary Gaussian processes
复制标题
平稳高斯过程非线性泛函的中心极限定理
DOI：
10.1007/bf01794427
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发表时间：
1989
影响因子：
2
通讯作者：
E. Slud
中科院分区：
文献类型：
--
作者：
D. W. Chambers;E. Slud
SummaryLet 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 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
$$Y_T (t) \equiv {{\int\limits_0^{Tt} {U_s \circ Yds} } \mathord{\left/ {\vphantom {{\int\limits_0^{Tt} {U_s \circ Yds} } {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right. \kern-\nulldelimiterspace} {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$
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).
SummaryLet 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 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
$$Y_T (t) \equiv {{\int\limits_0^{Tt} {U_s \circ Yds} } \mathord{\left/ {\vphantom {{\int\limits_0^{Tt} {U_s \circ Yds} } {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right. \kern-\nulldelimiterspace} {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$
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).