On exact asymptotics in the weak law of large numbers
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 803-813
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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.
@article{TVP_2004_49_4_a12,
author = {L. V. Rozovskii},
title = {On exact asymptotics in the weak law of large numbers},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {803--813},
publisher = {mathdoc},
volume = {49},
number = {4},
year = {2004},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2004_49_4_a12/}
}
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/