On the number of intersections of a level by a Gaussian stochastic process. II

Yu. K. Belyaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 444-457

Résumé

The main result of this paper which is a continuation of [8] is the following theorem: let $\xi_t$ be a stationary Gaussian process with $\mathbf M\xi_t=0$ and $\rho(t)$ be its correlation function. If $$ |\rho''(0)-\rho''(t)|\le\frac c{|\ln||t|^{1+\varepsilon}},\quad|t|\le t_0, $$ and $$ \rho(t)=o\biggl(\frac1{\ln t}\biggr),\quad\rho'(t)=o\biggl(\frac1{\sqrt{\ln t}}\biggr), $$ the moments of up-crossing of level $u$ form a Poisson random stream as $u\to\infty$. This result is a generalisation of a recent Cramer's theorem [10]. In the forthcoming third part of this investigation we'll consider other questions' about intersections by non-differentiable Gaussian processes.