Large Toeplitz operators and quadratic form generated by stationary Gaussian sequence

V. N. Solev ; L. Gerville-Reache

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 9, Tome 328 (2005), pp. 221-229

Résumé

Let $\Gamma_n(f,g)=\sum\limits_{-n\le t,\,s\le n}\,g_{t-s}X_tX_s$ – be a Toeplitz quadratic form generated by a real valued function $g(u)=\sum\limits_{-\infty}^{\infty}\,g_ke^{iku}$ and stationary sequence $X_n$ with spectral density $f$. Many sufficient conditions of asymptotic normality of the normalized quadratic form $\Psi_n(f,g)$ have been proposed since 1958. A less restrictive one was given in the paper of L. Giraitis and D. Surgailis (1990). Using a linear operator approach, we suggest a new vision of the problem and propose a new sufficient condition on the couple of functions $(f,g)$ even more effective.