This article is cited in 14 papers

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

Yu. K. Belyaev

Moscow

Abstract: 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.

Received: 17.05.1966

 English version: Theory of Probability and its Applications, 1967, 12:3, 392–404

Bibliographic databases:

• 
•