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Teor. Veroyatnost. i Primenen., 1967, Volume 12, Issue 3, Pages 444–457 (Mi tvp727)  

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}


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    This publication is cited in the following articles:
    1. D. G. Martirosyan, “Predelnaya teorema Puassona dlya chisla peresechenii nulevogo urovnya gaussovskogo statsionarnogo protsessa”, UMN, 27:3(165) (1972), 207–208  mathnet  mathscinet  zmath
    2. R. N. Miroshin, “On a Class of Multiple Integrals”, Math. Notes, 73:3 (2003), 359–369  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. 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  mathscinet  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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