On a Theorem of V. M. Zolotarev

Let ξ = η = ζ, where η and ζ are independent random variables, η has the probability density (7) and Eexp (ζ / 2) = K < ∞. It is shown that formula (10) is true if m 1, or if $0 < m < 1$ and condition (11) which is implied by (12) is satisfied. If P\ ζ < 0 \ = 0, inequality (13) holds for m 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 χ ² random variables, implies a result of V. M. Zolotarev [1].