Concerning a Certain Probability Problem

Abstract

Let $\xi _1 ,\xi _2 , \cdots $ be a sequence of independent $(0,1)$ normal random variables and let \[ \begin{gathered} \lambda _1^2 = \lambda _2^2 = \cdots \lambda _{n_1 }^2 , \hfill \\ \lambda _{n_1 + 1}^2 + \lambda _{n_1 + 2}^2 = \cdots = \lambda _{n_1 + n_2 }^2 , \hfill \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \hfill \\ \end{gathered} \] 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 $: \[ \begin{gathered} {\bf P}\{ \eta \geqq x\} = 1 - F_\eta (x) = \frac{K} {{\Gamma \left( {\frac{{n_1 }}{2}} \right)}}\left( {\frac{x}{{2\lambda _1^2 }}} \right)^{( n_1/ 2)-1} e^{ - x/2\lambda _1^2} [ 1 + \varepsilon _1 (x) ], \hfill \\ p_\eta (x)=\frac{K}{{\left( {2\lambda _1^2 } \right)^{n_1/ 2} \Gamma \left( {\frac{{n_1 }}{2}} \right)}}x^{\left( {{{h_1 } \mathord{\left/ {\vphantom {{h_1 } 2}} \right. \kern-\nulldelimiterspace} 2}} \right) - 1} e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} {2\lambda _1^2 \left( {1 + \varepsilon _2 (x)} \right)}}} \right. \kern-\nulldelimiterspace} {2\lambda _1^2}}} ( {1 + \varepsilon _2 (x)} ), \hfill \\ \end{gathered} \] where $\varepsilon _j (x) \to 0$ as $x \to \infty $ and \[ K = \prod\limits_{k = n_1 + 1}^\infty {\left( {1 - \frac{{\lambda _k^2 }}{{\lambda _1^2 }}} \right)^{ - 1} .} \]