## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 2 (1976), 326-334.

### Infinite $F$-Divisibility of Integer-Valued Random Variables

#### Abstract

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.

#### Article information

**Source**

Ann. Probab. Volume 4, Number 2 (1976), 326-334.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996138

**Digital Object Identifier**

doi:10.1214/aop/1176996138

**Mathematical Reviews number (MathSciNet)**

MR397830

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

**Keywords**

Dependence due to a constraint $F$-independence infinite divisibility bounded-sum variables binomial beta-binomial limits of sums of random variables

#### Citation

James, Ian R. Infinite $F$-Divisibility of Integer-Valued Random Variables. Ann. Probab. 4 (1976), no. 2, 326--334. doi:10.1214/aop/1176996138. https://projecteuclid.org/euclid.aop/1176996138