Partial divisibility of random sets

Abstract

In this article, we ask the following question: Let $V_X$ be the void functional of a random closed set $X$. For which $\alpha >0$ is $V_X^{\alpha }$ a void functional? We answer this question when $X$ is a random subset of a finite set. The result is then generalized to exponents which preserve complete monotonicity of functions on finite lattices. Also, we study the question of approximating an $m$-divisible random set by infinitely divisible random sets. We prove a theorem analogous to that of Arak’s classical result (Arak, 1981, 1982) on approximating an $m$-divisible random variable by infinitely divisible random variables.