## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 6 (1972), 2067-2071.

### On a Certain Class of Limit Distributions

#### Abstract

Suppose that $G$ is the distribution function $(\operatorname{df})$ of a (non-negative) $\mathrm{rv} Z$ satisfying the integral-functional equation $G(x) = b^{-1} \int^{lx}_0 \lbrack 1 - G(u)\rbrack du$, for $x > 0$, and zero for $x \leqq 0$, with $l \geqq 1$. Such a $\operatorname{df} G$ arises as the limit $\operatorname{df}$ of a sequence of iterated transformations of an arbitrary $\operatorname{df}$ of a $\operatorname{rv}$ having finite moments of all orders. When $l = 1, G$ must be the simple exponential $\operatorname{df}$ and is unique. It is shown, for $l > 1$, that there exists an infinite number of $\operatorname{df}$'s satisfying this equation. Using the fact that any $\operatorname{df} G$ which satisfies the given equation must have finite moments $\nu_k = K! b^kl^{k(k-1)/2}$ for $k = 0, 1, 2, \cdots$, it is shown that the $\operatorname{df}$ of the $\operatorname{rv} Z = UV$, where $U$ and $V$ are independent rv's having log-normal and simple exponential distributions, respectively, satisfies the integral functional equation. It is then easy to exhibit explicitly a family of solutions of the equation.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 6 (1972), 2067-2071.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177690886

**Digital Object Identifier**

doi:10.1214/aoms/1177690886

**Mathematical Reviews number (MathSciNet)**

MR353418

**Zentralblatt MATH identifier**

0253.60039

**JSTOR**

links.jstor.org

#### Citation

Shantaram, R.; Harkness, W. On a Certain Class of Limit Distributions. Ann. Math. Statist. 43 (1972), no. 6, 2067--2071. doi:10.1214/aoms/1177690886. https://projecteuclid.org/euclid.aoms/1177690886