On a Theorem of V. M. Zolotarev

Abstract

Let be a sequence of independent random variables, where has x ² distribution with n _r degrees of freedom and let be a strongly decreasing sequence of positive numbers such that. Then the random variable

$$ \xi = \sum\limits_{r = 1}^\infty {\sigma _r^2 \chi _r^2 } $$

((1))

exists with probability 1. V. M. Zolotarev [1] has shown that

$$\begin{array}{*{20}{c}} {\lim } \\ {x \to \infty } \\ \end{array} \frac{{{{\mathcal{P}}_{\xi }}(x)}}{{\mathcal{P}\sigma _{1}^{2}x_{1}^{2}\left( x \right)}} = {{\prod\limits_{{r = 2}}^{\infty } {\left( {1 - \frac{{\sigma _{r}^{2}}}{{\sigma _{r}^{2}}}} \right)} }^{{ - {{n}_{r}}/2}}}$$

((2))

where Pξ′(x) is the probability density of the random variable ξ′. From (2) one can easily obtain an asymptotic expression for P{ξ > x} for x → ∞.