On a Theorem of V. M. Zolotarev

Abstract

Let $\xi = \eta = \zeta $, where $\eta $ and $\zeta $ are independent random variables, $\eta $ has the probability density (7) and ${\bf E}\exp ({\zeta / 2}) = K < \infty $. It is shown that formula (10) is true if $m \geqq 1$, or if $0 < m < 1$ and condition (11) which is implied by (12) is satisfied. If ${\bf P}\{ {\zeta < 0} \} = 0$, inequality (13) holds for $m \geqq 1$. Formula (14) is true if conditions (15) and (in the case $r > m - 1$) (16) are satisfied. An application to the random variable (1), a weighted sum of independent $\chi ^2 $ random variables, implies a result of V. M. Zolotarev [1].