On Exact Asymptotics in the Weak Law of Large Numbers for Sums of Independent Random Variables with a Common Distribution Function from the Domain of Attraction of a Stable Law

Abstract

Let us consider independent identically distributed random variables $X_1, X_2, \dots\,$, such that $$ U_n=\frac{S_n}{B_n} -n\,a_n \longrightarrow \xi_\alpha\qq \mbox{weakly as}\quad n\to\infty, $$ where $S_n = X_1 + \cdots + X_n$, $B_n>0$, $a_n$ are some numbers $(n\geq 1)$, and a random variable $\xi_\alpha$ has a stable distribution with characteristic exponent $\alpha\in (0, 2)$. Our basic purpose is to find conditions under which $$ \sum_n f_n {\bf P}\{|U_n|\geq\varepsilon\varphi_n\}\sim \sum_n f_n {\bf P}\{|\xi_\al|\ge\varepsilon\varphi_n\},\qquad\varepsilon\searrow 0, $$ with a positive sequence $\varphi_n$, which tends to infinity and satisfies mild additional restrictions, and with a nonnegative sequence $f_n$ such that $\sum_n f_n =\infty $.