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This article is cited in 9 scientific papers (total in 9 papers)
Short Communications
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
L. V. Rozovskii Saint-Petersburg Chemical-Pharmaceutical Academy
Abstract:
Let us consider independent identically distributed random variables
$X_1, X_2, \ldots\,$, 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 $\alpha\in (0, 2)$.
$$
\sum_n f_nP\{|U_n|\geq\varepsilon\varphi_n\}\sim
\sum_n f_nP\{|\xi_\alpha|\ge\varepsilon\varphi_n\},\qquad\varepsilon\searrow 0,
$$
Our basic purpose is to find conditions under which
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.
Received: 20.11.2002
Citation:
L. V. Rozovskii, “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”, Teor. Veroyatnost. i Primenen., 48:3 (2003), 589–596; Theory Probab. Appl., 48:3 (2004), 561–568
Linking options:
https://www.mathnet.ru/eng/tvp273https://doi.org/10.4213/tvp273 https://www.mathnet.ru/eng/tvp/v48/i3/p589
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