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Teoriya Veroyatnostei i ee Primeneniya, 1970, Volume 15, Issue 1, Pages 116–119 (Mi tvp1569)  

Short Communications

Carleman's classes for stationary processes

S. A. Ivankov

Moscow
Abstract: The main result of the present paper consists in demonstration of the fact that sample functions of a stationary stochastic process belong, with probability one, to Carleman's class $С\{m_n\}$, if the correlation function of the process belongs to the same class, and if
$$ 0<D=\inf_y\biggl\{y\colon\varlimsup_{n\to\infty}\frac{m_{2n}}{m^2_ny^{2n}}=0\biggr\}<\infty $$

For processes satisfying the conditions
$$ \varliminf_{n\to\infty}\mathbf P\{(\xi^{(n)}(0))^2>\mathbf M(\xi^{(n)}(0))^2\}>0,\quad1<\frac{m_n}{m_{n-1}}<Vn^w, $$
where $V$ and $w$ are positive constants, the converse assertion is proved to be also true.
Received: 28.03.1968
English version:
Theory of Probability and its Applications, 1970, Volume 15, Issue 1, Pages 115–117
DOI: https://doi.org/10.1137/1115010
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: S. A. Ivankov, “Carleman's classes for stationary processes”, Teor. Veroyatnost. i Primenen., 15:1 (1970), 116–119; Theory Probab. Appl., 15:1 (1970), 115–117
Citation in format AMSBIB
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\by S.~A.~Ivankov
\paper Carleman's classes for stationary processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1970
\vol 15
\issue 1
\pages 116--119
\mathnet{http://mi.mathnet.ru/tvp1569}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=288836}
\zmath{https://zbmath.org/?q=an:0216.46703|0196.18703}
\transl
\jour Theory Probab. Appl.
\yr 1970
\vol 15
\issue 1
\pages 115--117
\crossref{https://doi.org/10.1137/1115010}
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