Asymptotics of large deviations of

V. R. Fatalov

Sbornik. Mathematics, Tome 194 (2003) no. 3, pp. 369-390

Résumé

A general result is obtained on exact asymptotics of the probabilities $$ \mathsf P\biggl\{\int_0^1|\xi(t)|^p\,dt>u^p\biggr\} $$ as $u\to\infty$ and $p>0$ for Gaussian processes $\xi(t)$. The general theorem is applied for the calculation of these asymptotics in the cases of the following processes: the Wiener process $w(t)$, the Brownian bridge, and the stationary Gaussian process $\eta(t):=w(t+1)-w(t)$, $t\in\mathbb R^1$. The Laplace method in Banach spaces is used. The calculations of the constants reduce to solving an extremum problem for the action functional and studying the spectrum of a differential operator of the second order of Sturm–Liouville type.