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June, 1980 Domains of Partial Attraction and Tightness Conditions
Naresh C. Jain, Steven Orey
Ann. Probab. 8(3): 584-599 (June, 1980). DOI: 10.1214/aop/1176994728


Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed, random variables with a common distribution function $F. S_n$ denotes $X_1 + \cdots + X_n$. An increasing sequence of positive integers $(n_i)$ is defined to belong to $\mathscr{N}(F)$ if there exist normalizing sequences $(b_k)$ and $(a_k)$, with $a_k \rightarrow \infty$, so that every subsequence of $(a^{-1}_{n_i} S_{n_i} - b_{n_i})$ has a further subsequence converging in distribution to a nondegenerate limit. The main concern here is a description of $\mathscr{N}(F)$ in terms of $F$. This includes also conditions for $\mathscr{N}(F)$ to be void, as well as for $(1, 2, \cdots)\in \mathscr{N}(F)$, thus improving on some classical results of Doeblin. It is also shown that if there exists a unique type of laws so that $F$ is in the domain of partial attraction of a probability law if and only if the law belongs to that type, then in fact $F$ is in the domain of attraction of these laws.


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Naresh C. Jain. Steven Orey. "Domains of Partial Attraction and Tightness Conditions." Ann. Probab. 8 (3) 584 - 599, June, 1980.


Published: June, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0442.60050
MathSciNet: MR573294
Digital Object Identifier: 10.1214/aop/1176994728

Primary: 60G50

Keywords: domain of attraction , domain of partial attraction , set of uniform decrease , Sums of independent random variables , tight sequence

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 3 • June, 1980
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