Concerning a Certain Probability Problem

V. M. Zolotarev

Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 2, pp. 219-222

Résumé

Let $\xi_1,\xi_2,\dots$ 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 $$\lambda_1^2>\lambda_{n_1+1}^2>\cdots{\text{and}}\sum\limits_k\lambda_k^2<\infty.$$ We prove the following asymptotic formula for the distribution of the random variable $\eta =\sum\nolimits_k {\lambda_k^2}\xi_k^2$: $$\mathbf P\{\eta\geq x\}=1-F_\eta(x)=\frac{K}{\Gamma\left(\frac{n_1}2\right)}\left( \frac{x}{2\lambda_1^2}\right)^{(n_1/2)-1}e^{-x/2\lambda_1^2}[1+\varepsilon_1(x)],\\ p_\eta(x)=\frac{K}{{\left({2\lambda_1^2}\right)^{n_1/2}\Gamma\left({\frac{{n_1}}2}\right)}}x^{\left({{{h_1}{\left/{\vphantom{{h_1}2}}\right.}2}}\right)-1}e^{{{-x}{\left/{\vphantom{{-x}{2\lambda_1^2\left({1+\varepsilon_2 (x)}\right)}}}\right.}{2\lambda_1^2}}}({1+\varepsilon_2(x)}),$$ where $\varepsilon_j(x)\to 0$ as $x\to\infty$ and $$K=\prod\limits_{k=n_1+1}^\infty{\left({1-\frac{{\lambda_k^2}}{{\lambda_1^2}}}\right)^{-1}}.$$