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Asymptotic Normality of Cross-correlogram Estimates of the Response Function
Authors:Buldygin  Valery  Utzet  Frederic  Zaiats  Vladimir
Affiliation:(1) Department of Higher Mathematics No. 1, National Technical University of Ukraine "ldquo"Kyïv Polytechnic Institute,"rdquo" prosp, Peremohy 37, 02056 Kyïv, Ukraine
Abstract:The problem of estimation of an unknown response function of a time-invariant continuous linear system is considered. Discrete-time sample input–output cross-correlograms are taken as estimates of the response function. The inputs are supposed to be zero-mean stationary Gaussian processes close, in some sense, to a white noise. Both asymptotic normality of finite-dimensional distributions of the estimates and their asymptotic normality in spaces of continuous functions are studied. Our basic tool is a new integral representation for cumulants of the estimate as a finite sum of integrals involving cyclic products of kernels. Some inequalities for these integrals are obtained and their asymptotic behaviour is studied.
Keywords:response function  cross-correlogram  estimate  integral involving a cyclic product of kernels  asymptotic normality
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