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April, 1976 Infinite $F$-Divisibility of Integer-Valued Random Variables
Ian R. James
Ann. Probab. 4(2): 326-334 (April, 1976). DOI: 10.1214/aop/1176996138


Consider $m \geqq 2$ nonnegative, integer-valued random variables $X_1^{(n)},\cdots, X_m^{(n)}$ satisfying $\sum^m_{j=1} X_j^{(n)} \leqq n$. If $(X_1^{(n)},\cdots, X_m^{(n)})$ is one member of a family of random vectors, indexed by different values of the bound $n$, Darroch has proposed a definition of "independence except for the constraint," termed $F$-independence, which relates members of the family through their conditional distributions. In this paper we study the limit theory for sums of nonnegative, integer-valued variables, when the sums are bounded and the variables $F$-independent. The $F$-independence analogue of infinite divisibility, termed infinite $F$-divisibility, is defined and characterized, and it is shown that limit distributions of sums of $F$-independent, asymptotically negligible variables are infinitely $F$-divisible. Conditions under which the limit is binomial are given. Our results apply to families of random variables, induced by the $F$-independence definition, and their role in the theory is discussed.


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Ian R. James. "Infinite $F$-Divisibility of Integer-Valued Random Variables." Ann. Probab. 4 (2) 326 - 334, April, 1976.


Published: April, 1976
First available in Project Euclid: 19 April 2007

MathSciNet: MR397830
Digital Object Identifier: 10.1214/aop/1176996138

Primary: 60G50

Keywords: $F$-independence , beta-binomial , binomial , bounded-sum variables , Dependence due to a constraint , Infinite divisibility , limits of sums of random variables

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 2 • April, 1976
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