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Teor. Veroyatnost. i Primenen., 1961, Volume 6, Issue 2, Pages 219–222 (Mi tvp4769)  

This article is cited in 6 scientific papers (total in 6 papers)

Short Communications

Concerning a Certain Probability Problem

V. M. Zolotarev

Moscow

Abstract: Let $\xi_1,\xi_2,…$ be a sequence of independent $(0,1)$ normal random variables and let
$$\lambda_1^2=\lambda_2^2=\cdots\lambda_{n_1}^2,l
\lambda_{n_1+1}^2+\lambda_{n_1+2}^2=\cdots=\lambda_{n_1+n_2}^2,
\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$$
be a sequence of positive numbers such that
$$\lambda_1^2>\lambda_{n_1+1}^2>\cdots{and}\sum\limits_k\lambda_k^2<\infty.$$

We prove the following asymptotic formula for the distribution of the random variable $\eta =\sum\nolimits_k {\lambda_k^2}\xi_k^2$:
$$\mathbf P\{\eta\geq x\}=1-F_\eta(x)=\frac{K}{\Gamma(\frac{n_1}2)}( \frac{x}{2\lambda_1^2})^{(n_1/2)-1}e^{-x/2\lambda_1^2}[1+\varepsilon_1(x)], p_\eta(x)=\frac{K}{{({2\lambda_1^2})^{n_1/2}\Gamma({\frac{{n_1}}2})}}x^{({{{h_1}{/{\vphantom{{h_1}2}}.}2}})-1}e^{{{-x}{/{\vphantom{{-x}{2\lambda_1^2({1+\varepsilon_2 (x)})}}}.}{2\lambda_1^2}}}({1+\varepsilon_2(x)}),$$
where $\varepsilon_j(x)\to 0$ as $x\to\infty$ and
$$K=\prod\limits_{k=n_1+1}^\infty{({1-\frac{{\lambda_k^2}}{{\lambda_1^2}}})^{-1}}.$$


Full text: PDF file (380 kB)

English version:
Theory of Probability and its Applications, 1961, 6:2, 201–204

Received: 22.12.1960

Citation: V. M. Zolotarev, “Concerning a Certain Probability Problem”, Teor. Veroyatnost. i Primenen., 6:2 (1961), 219–222; Theory Probab. Appl., 6:2 (1961), 201–204

Citation in format AMSBIB
\Bibitem{Zol61}
\by V.~M.~Zolotarev
\paper Concerning a Certain Probability Problem
\jour Teor. Veroyatnost. i Primenen.
\yr 1961
\vol 6
\issue 2
\pages 219--222
\mathnet{http://mi.mathnet.ru/tvp4769}
\transl
\jour Theory Probab. Appl.
\yr 1961
\vol 6
\issue 2
\pages 201--204
\crossref{https://doi.org/10.1137/1106025}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Piterbarg, V. R. Fatalov, “The Laplace method for probability measures in Banach spaces”, Russian Math. Surveys, 50:6 (1995), 1151–1239  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. M. S. Ermakov, “On large deviations of type II error probabilities of Kolmogorov and omega-squared tests”, J. Math. Sci. (N. Y.), 128:1 (2005), 2538–2555  mathnet  crossref  mathscinet  zmath
    3. V. R. Fatalov, “Exact Asymptotics of Large Deviations of Stationary Ornstein–Uhlenbeck Processes for $L^p$-Functional, $p>0$”, Problems Inform. Transmission, 42:1 (2006), 46–63  mathnet  crossref  mathscinet  zmath  elib  elib
    4. V. R. Fatalov, “Some asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 157:2 (2008), 1606–1625  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. V. R. Fatalov, “On the Laplace method for Gaussian measures in a Banach space”, Theory Probab. Appl., 58:2 (2014), 216–241  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. Gao FuChang Ya.X., “Upper Tail Probabilities of Integrated Brownian Motions”, Sci. China-Math., 58:5 (2015), 1091–1100  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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