Correlogram estimation of response functions of linear systems in scheme of some independent samples

The problem of estimation of an unknown response function from $L_{2}(\mathbf{R})$ of a linear system is considered. The inputs are supposed to be stationary zero-mean Gaussian almost surely sample continuous processes. We take the integral-type sample input-output cross-correlograms as estimators of the response function and apply the scheme of some independent samples, when the pair of inputs and outputs are observed. The asymptotic normality of the distributions of centered cross-correlogram estimations in the space of continuous functions and the construction of the confidence bands for the limiting process are discussed.