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 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 3, Pages 589–596 (Mi tvp273)

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

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.

DOI: https://doi.org/10.4213/tvp273

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English version:
Theory of Probability and its Applications, 2004, 48:3, 561–568

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”, Teor. Veroyatnost. i Primenen., 48:3 (2003), 589–596; Theory Probab. Appl., 48:3 (2004), 561–568

Citation in format AMSBIB
\Bibitem{Roz03} \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 \jour Teor. Veroyatnost. i Primenen. \yr 2003 \vol 48 \issue 3 \pages 589--596 \mathnet{http://mi.mathnet.ru/tvp273} \crossref{https://doi.org/10.4213/tvp273} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2141353} \zmath{https://zbmath.org/?q=an:1054.60031} \transl \jour Theory Probab. Appl. \yr 2004 \vol 48 \issue 3 \pages 561--568 \crossref{https://doi.org/10.1137/S0040585X97980592} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000224300900015} 

• http://mi.mathnet.ru/eng/tvp273
• https://doi.org/10.4213/tvp273
• http://mi.mathnet.ru/eng/tvp/v48/i3/p589

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This publication is cited in the following articles:
1. 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”, Theory Probab. Appl., 49:4 (2005), 724–734
2. Spătaru A., “Precise asymptotics for a series of T. L. Lai”, Proc. Amer. Math. Soc., 132:11 (2004), 3387–3395
3. Huang Wei, Zhang Lixin, “Precise rates in the law of the logarithm in the Hilbert space”, J. Math. Anal. Appl., 304:2 (2005), 734–758
4. V. V. Buldygin, O. I. Klesov, J. G. Steinebach, “Precise asymptotics over a small parameter for a series of large deviation probabilities”, Theory Stoch. Process., 13(29):1 (2007), 44–56
5. L. V. Rozovskii, “Small deviations of modified sums of independent random variables”, J. Math. Sci. (N. Y.), 159:3 (2009), 341–349
6. L. V. Rozovskii, “Probabilities of small deviations of the maximum of partial sums”, Theory Probab. Appl., 54:4 (2010), 717–724
7. Theory Probab. Appl., 54:4 (2010), 703–717
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