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

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

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} 

• http://mi.mathnet.ru/eng/tvp4769
• http://mi.mathnet.ru/eng/tvp/v6/i2/p219

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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
4. V. R. Fatalov, “Some asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 157:2 (2008), 1606–1625
5. V. R. Fatalov, “On the Laplace method for Gaussian measures in a Banach space”, Theory Probab. Appl., 58:2 (2014), 216–241
6. Gao FuChang Ya.X., “Upper Tail Probabilities of Integrated Brownian Motions”, Sci. China-Math., 58:5 (2015), 1091–1100