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 Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 4, Pages 803–813 (Mi tvp198)

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. II

L. V. Rozovskii

Abstract: Let us consider independent identically distributed random variables $X_1, X_2, …$, 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
$$\sum_n f_n{P}\{U_n\geq\varepsilon\varphi_n\}\sim \sum_n f_n{P}\{\xi_\alpha\ge\varepsilon\varphi_n\}, \qquad\varepsilon\searrow 0,$$
with a positive sequence $\varphi_n$, which tends to infinity and satisfies mild additional restrictions, and with a nonnegative sequence $f_n$ such that $\sum_n f_n =\infty$.

Keywords: independent random variables, law of large numbers, stable law.

DOI: https://doi.org/10.4213/tvp198

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English version:
Theory of Probability and its Applications, 2005, 49:4, 724–734

Bibliographic databases:

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. II”, Teor. Veroyatnost. i Primenen., 49:4 (2004), 803–813; Theory Probab. Appl., 49:4 (2005), 724–734

Citation in format AMSBIB
\Bibitem{Roz04} \by L.~V.~Rozovskii \paper 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.~II \jour Teor. Veroyatnost. i Primenen. \yr 2004 \vol 49 \issue 4 \pages 803--813 \mathnet{http://mi.mathnet.ru/tvp198} \crossref{https://doi.org/10.4213/tvp198} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2144252} \zmath{https://zbmath.org/?q=an:1103.60025} \transl \jour Theory Probab. Appl. \yr 2005 \vol 49 \issue 4 \pages 724--734 \crossref{https://doi.org/10.1137/S0040585X97981408} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000234407500013} 

• http://mi.mathnet.ru/eng/tvp198
• https://doi.org/10.4213/tvp198
• http://mi.mathnet.ru/eng/tvp/v49/i4/p803

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This publication is cited in the following articles:
1. L. V. Rozovskii, “Small deviations of modified sums of independent random variables”, J. Math. Sci. (N. Y.), 159:3 (2009), 341–349
2. L. V. Rozovskii, “Probabilities of small deviations of the maximum of partial sums”, Theory Probab. Appl., 54:4 (2010), 717–724
3. Theory Probab. Appl., 54:4 (2010), 703–717
4. Rozovsky L., “Super large deviation probabilities for sums of independent lattice random variables with exponential decreasing tails”, Statistics & Probability Letters, 82:1 (2012), 72–76
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