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.