On the Asymptotic Behaviour of the Estimate of the Spectral Function for a Stationary Gaussian Process

T. L. Malevič

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 386-390

Résumé

Let $\xi _n$, $n=0$, $\pm 1,\pm 2,\dots$, be a real stationary Gaussian sequence with an absolutely continuous spectral function $F(\lambda)$, and let $F_N (\lambda)$ be the sample spectral function.We assume that $F(\lambda)$ has no interval of constancy, and $f(\lambda)=F'(\lambda)\in L_2[0,\pi]$. Then the sequence of measures $P_N$ generated by the process $\zeta_N(\lambda)=\sqrt N[F_n(\lambda)-F(\lambda)]$ converges weakly to the measure which is generated by the Gaussian process $\zeta(\lambda)$ with ${\mathbf M}\zeta(\lambda)=0$ and $$ {\mathbf M}\zeta(\lambda)\zeta(\mu)=2\pi\int_0^{\min(\lambda\mu)}f^2(x)\,dx. $$ A similar result holds for the process $\xi_t$ with continuous time, $0\leqslant t<+\infty$.