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Central limit theorems for nonlinear functionals of stationary Gaussian processes

Authors:

Affiliation:

1. Department of Mathematics, Boston College, 02167, Chestnut Hill, MA, USA
2. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

Abstract:

Let X=(X_t,t isin

) be a stationary Gaussian process on ( OHgr

,P), letH(X) be the Hilbert space of variables inL² ( OHgr

,P) which are measurable with respect toX, and let (U_s,s isin

) be the associated family of time-shift operators. We sayY isin

H(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively,], where

$Y_T (t) equiv {{intlimits_0^{Tt} {U_s circ Yds} } mathord{left/ {vphantom {{intlimits_0^{Tt} {U_s circ Yds} } {left{ {Varleft( {intlimits_0^T {U_s circ Yds} } right)} right}^{{1 mathord{left/ {vphantom {1 2}} right. kern-nulldelimiterspace} 2}} }}} right. kern-nulldelimiterspace} {left{ {Varleft( {intlimits_0^T {U_s circ Yds} } right)} right}^{{1 mathord{left/ {vphantom {1 2}} right. kern-nulldelimiterspace} 2}} }}$

Keywords:
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