Short Communications

О некоторых свойствах сопровождающих законов для симметричных функций распределений

Yu. P. Studnev

Uzhgorod

Abstract: Let $\{\xi_k\}$ be a sequence of independent random variables with the same symmetric distribution function $F(x)$ which has a non-negative characteristic function and $F_n(x)$ be the distribution function of the sum $s_n=\xi_1+\dots+\xi_n$. Denote by $\mathfrak G$ the set of infinitely divisible laws.
In the paper we show by elementary methods that there exist such metrics
$$ \rho_i(F_n,G)\quad(G\in\mathfrak G),\quad i=1,2,\dots, $$
invariant with respect, to linear transformations of the arguments, that the inequality
$$ \inf_{G\in\mathfrak G}\rho_i(F_n,G)\le Cn^{-1} $$
where $C$ is an absolute constant, holds.

Received: 14.06.1966

 English version: Theory of Probability and its Applications, 1968, 13:4, 701–703

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