Asymptotic behavior of the log-likelihood function when the spectral function has polynomial zeros

M. S. Ginovyan

Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part V, Tome 108 (1981), pp. 5-21

Résumé

The Gaussian stationary process $x_t$, $t=0,\pm1,\dots$ with zero mean spectral dencity $f$: $$ f(\lambda)=|Q_m(e^{i\lambda})|^2h(\lambda), $$ where $Q_m(z)$ is polynomial of degree $m$ with roots on the unit circle is, considered. The purpose of this paper is to investigate the asymptotic behavior of the logarithm of likelihood function $\mathscr L_n$. We show, that under the suitable condition on the spectral density $f$ the simple approximation $\widetilde{\mathscr L}_n$ of the function $\mathscr L_n$ satisfying the condition $$ \frac1{\sqrt n}(\mathscr L_n-\widetilde{\mathscr L}_n)\to0\text{ when }n\to\infty $$ by probability exist.