Geodesic
On exact asymptotics in the weak law of large numbers
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 803-813 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let us consider independent identically distributed random variables $X_1, X_2, \dots\,$, such that $$ U_n=\frac{S_n}{B_n} -n\,a_n \longrightarrow \xi_\alpha\qquad weakly as\quad n\to\infty, $$ 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 $\alpha\in[1,2]$. Our basic purpose is to find conditions under which $$ \sum_n f_n{P}\big\{U_n\geq\varepsilon\varphi_n\big\}\sim \sum_n f_n{P}\big\{\xi_\alpha\ge\varepsilon\varphi_n\big\}, \qquad\varepsilon\searrow 0, $$ 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 $.
Keywords: independent random variables, law of large numbers, stable law. 
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L. V. Rozovskii. On exact asymptotics in the weak law of large numbers. Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 803-813. http://geodesic.mathdoc.fr/item/TVP_2004_49_4_a12/

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