|
This article is cited in 3 scientific papers (total in 3 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
$$
\rho(t)=o(\frac1{\ln t}),\quad\rho'(t)=o(\frac1{\sqrt{\ln t}}),
$$
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.
Full text:
PDF file (729 kB)
English version:
Theory of Probability and its Applications, 1967, 12:3, 392–404
Bibliographic databases:
Received: 17.05.1966
Citation:
Yu. K. Belyaev, “On the number of intersections of a level by a Gaussian stochastic process. II”, Teor. Veroyatnost. i Primenen., 12:3 (1967), 444–457; Theory Probab. Appl., 12:3 (1967), 392–404
Citation in format AMSBIB
\Bibitem{Bel67}
\by Yu.~K.~Belyaev
\paper On the number of intersections of a~level by a~Gaussian stochastic process.~II
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 3
\pages 444--457
\mathnet{http://mi.mathnet.ru/tvp727}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=217859}
\zmath{https://zbmath.org/?q=an:0162.21201}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 3
\pages 392--404
\crossref{https://doi.org/10.1137/1112051}
Linking options:
http://mi.mathnet.ru/eng/tvp727 http://mi.mathnet.ru/eng/tvp/v12/i3/p444
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Cycle of papers
This publication is cited in the following articles:
-
D. G. Martirosyan, “Predelnaya teorema Puassona dlya chisla peresechenii nulevogo urovnya gaussovskogo statsionarnogo protsessa”, UMN, 27:3(165) (1972), 207–208
-
R. N. Miroshin, “On a Class of Multiple Integrals”, Math. Notes, 73:3 (2003), 359–369
-
Zakharov V A., Chernoyarov V O., Salnikova V A., Faulgaber A.N., “The Distribution of the Absolute Maximum of the Discontinuous Stationary Random Process With Raileigh and Gaussian Components”, Eng. Lett., 27:1 (2019), 53–65
|
Number of views: |
This page: | 236 | Full text: | 106 |
|