On the second moments of an estimate of the spectral function

M. P. Shaifer

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 536-539

Résumé

A real stationary stochastic process $\{x_n\}$, $x_n=\sum_{k=-\infty}^\infty a_k\xi_{k+n}$ where $\xi_k$ are equally distributed independent random variables with $\mathbf E\xi_0=0$, $\mathbf E\xi_0^2=1$, $\mathbf E\xi_0^4<\infty$ and $\sum_{k=-\infty}^\infty a_k^2<\infty$ is considered. The asymptotic properties of the expression $$ \operatorname{cov}\biggl(\int_{-\pi}^\pi T_1(\lambda)Y_N(\lambda)\,d\lambda,\ \int_{-\pi}^\pi T_2(\lambda)Y_N(\lambda)\,d\lambda\biggr) $$ where $$ Y_N(\lambda)=\frac1{2\pi N}\biggl|\sum_{j=1}^Nx_je^{i\lambda j}\biggr|^2 $$ and $\operatorname{Var}T_i(\lambda)<\infty$ ($i=1,2$) are investigated.