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May, 1982 Limit Points of $\{n^{-1/\alpha}S_n\}$
Joop Mijnheer
Ann. Probab. 10(2): 382-395 (May, 1982). DOI: 10.1214/aop/1176993864


Let $X_1, X_2, \cdots$ be a sequence of independent identically distributed (i.i.d.) positive random variables in the domain of attraction of a completely asymmetric stable law with characteristic exponent $\alpha \in (0, 1)$, i.e. their common distribution function $G$ is given by $P(X_1 > x) = 1 - G(x) = x^{-\alpha}L(x),$ where $L$ is a slowly varying function at infinity. In this paper we study the set of limit points of $\{n^{-1/\alpha}(X_1 + \cdots + X_n): n = 1,2, \cdots\}$. The sets of limit points that are possible are $\{0\}, \{\infty\}, \lbrack 0, \infty\rbrack$ and $\lbrack b, \infty\rbrack$ for some positive number $b$. In Section 2 we consider the case where $L$ is non-decreasing and in Section 3 the case where $L$ is non-increasing. In both sections we give the conditions in terms of $L$ for each of the limit sets.


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Joop Mijnheer. "Limit Points of $\{n^{-1/\alpha}S_n\}$." Ann. Probab. 10 (2) 382 - 395, May, 1982.


Published: May, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0478.60057
MathSciNet: MR647511
Digital Object Identifier: 10.1214/aop/1176993864

Primary: 60G50
Secondary: 60F15 , 60J30

Keywords: limit points , Normed sums of independent random variables , Stable process

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • May, 1982
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