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Teoriya Veroyatnostei i ee Primeneniya, 1971, Volume 16, Issue 2, Pages 353–360 (Mi tvp2228)  

Short Communications

On the Tchebyshev inequality ih the two-dimensional case

L. V. Arharov

Moscow
Abstract: Let $\xi_1$, $\xi_2$ be indepent random variables satisfying the conditions
$$ \mathbf P\{\xi_1\ge0\}=\mathbf P\{\xi_2\ge0\}=1\quad\mathbf M\xi_1=\mathbf M\xi_2=1. $$
For positive $\Delta_1$ and $\Delta_2$, the inequality
\begin{gather*} \mathbf P\{\Delta_1\min(\xi_1,\xi_2)+\Delta_2\max(\xi_1,\xi_2)\ge c\}\le \\ \le\max[(\Delta_1+\Delta_2)^2,\Delta_2/(1-\Delta_1),2\Delta_2(1-\Delta_2)+\Delta_2^2] \end{gather*}
is proved. Moreover, if $\xi_1$ and $\xi_2$ are equally distributed, then it is proved that
$$ \mathbf P\{\Delta_1\min(\xi_1,\xi_2)+\Delta_2\max(\xi_1,\xi_2)\ge c\}\le\max[(\Delta_1+\Delta_2)^2;2\Delta_2(1-\Delta_2)+\Delta_2^2]. $$
Received: 04.09.1969
English version:
Theory of Probability and its Applications, 1971, Volume 16, Issue 2, Pages 356–361
DOI: https://doi.org/10.1137/1116034
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: L. V. Arharov, “On the Tchebyshev inequality ih the two-dimensional case”, Teor. Veroyatnost. i Primenen., 16:2 (1971), 353–360; Theory Probab. Appl., 16:2 (1971), 356–361
Citation in format AMSBIB
\Bibitem{Ark71}
\by L.~V.~Arharov
\paper On the Tchebyshev inequality ih the two-dimensional case
\jour Teor. Veroyatnost. i Primenen.
\yr 1971
\vol 16
\issue 2
\pages 353--360
\mathnet{http://mi.mathnet.ru/tvp2228}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=285040}
\zmath{https://zbmath.org/?q=an:0239.60022}
\transl
\jour Theory Probab. Appl.
\yr 1971
\vol 16
\issue 2
\pages 356--361
\crossref{https://doi.org/10.1137/1116034}
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