On Toeplitz type quadratic functionals of stationary Gaussian processes (Q1343609)

Let \(X(t)\) be a stationary, centered Gaussian process with spectral density \(f\) and let \(L_ T = {1 \over T} \int^ T_ 0 \int^ T_ 0 \widehat g(t -s) X(t) X(s) dt ds\) be a Toeplitz-type quadratic functional of \(X\), where \(\widehat g\) is the Fourier transform of some real, even, integrable function \(g\). A CLT for \(L_ T\) is proved. Namely, if \(f \in L_ p\) and \(g \in L_ q\), \(1/p + 1/q \leq 1/2\), then \(\sqrt T(L_ T - EL_ T)\) converges in distribution to the normal distribution with mean 0 and variance \(\| fg \|^ 2_ 2\). This generalizes the result of \textit{F. Avram} [ibid. 79, No. 1, 37-45 (1988; Zbl 0648.60043)] for discrete time processes. The result is applied to the estimation of linear functionals of an unknown spectral density, based on the observation of \(X\) on \([0,T]\). Upper bounds for the minimax mean square risk of estimators, being Toeplitz-type quadratic functionals, are obtained. They are similar to those by \textit{I. A. Ibragimov} and \textit{R. Z. Khas'minskij} [J. Sov. Math. 44, No. 4, 454-465 (1989); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 153, 45-59 (1986; Zbl 0616.62030)] for probability density functions.