Asymptotic properties of distributions of the maximum of a Gaussian nonstationary process occurring in covariance statistic

E. I. Ostrovskii ; s. Yu. Tsykunova

Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 3, pp. 641-649

Résumé

The paper considers a separable Gaussian centered process $\eta (t)$ with a covariance function of the type $$ \mathbf{M}\eta(t)\eta(s)=4\pi\int_{-\infty}^{+\infty}\cos\lambda t\cos\lambda sf^2(\lambda)\,d\lambda $$ for different restrictions on the spectral density $f(\lambda )$.Such processes appear as weak limits of normed deviations of empirical covariance function $$ \eta(t)=\lim_{T\to\infty}\sqrt T(r_T(t)-r(t)) $$ as $T\to\infty$. Here $f(\lambda)=(2\pi)^{-1}\int\exp(-i\lambda t)r(t)\,dt$. The paper studies the asymptotic behavior of a probability $$ P(u,s)=\mathbf{P}\biggl(\sup_{|t|}|\eta(t)|>u\biggr) $$ (as $u\to\infty$). Either the exact asymptotic or upper and lower estimates differing by a multiplicative constant are obtained for this probability. The case of Gaussian centered separable field is also considered. The results obtained may be applied for constructing the confidence interval for $r(t)$ in the uniform norm.

Détail

Keywords: covariance function, exact asymptotic, spectral density, separable centered Gaussian field, Talagrand theorem.