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Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 1, Pages 147–150 (Mi tvp4196)  

Short Communications

Some Properties of the Supremum of Sums of Stationary Related Random Variables

A. A. Borovkov

Novosibirsk

Abstract: Let $\{\xi_j, -\infty<j<\infty\}$ be a strong-sense stationary sequence
$$ X_k=\sum_{j=1}^k \xi_j,\quad X_0=0,\quad \eta=\sup_{k\ge 0}X_k,\quad \theta=\inf_{k\ge 0}X_k. $$

We prove two theorems; the first explains the connection between the nature of $\{\xi_j\}$ and the distributions of $\eta$ and $\theta$; the second gives a useful inequality for $\mathbf{P}(\eta>0)$ in terms of the distribution of $\xi_j$.

Full text: PDF file (577 kB)

English version:
Theory of Probability and its Applications, 1972, 17:1, 149–151

Bibliographic databases:

Received: 02.03.1971

Citation: A. A. Borovkov, “Some Properties of the Supremum of Sums of Stationary Related Random Variables”, Teor. Veroyatnost. i Primenen., 17:1 (1972), 147–150; Theory Probab. Appl., 17:1 (1972), 149–151

Citation in format AMSBIB
\Bibitem{Bor72}
\by A.~A.~Borovkov
\paper Some Properties of the Supremum of Sums of Stationary Related Random Variables
\jour Teor. Veroyatnost. i Primenen.
\yr 1972
\vol 17
\issue 1
\pages 147--150
\mathnet{http://mi.mathnet.ru/tvp4196}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=315776}
\zmath{https://zbmath.org/?q=an:0267.60030}
\transl
\jour Theory Probab. Appl.
\yr 1972
\vol 17
\issue 1
\pages 149--151
\crossref{https://doi.org/10.1137/1117012}


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