Partial divisibility of random sets and powers of completely monotone functions

In this article, we study exponents which preserve complete monotonicity of functions on lattices. We prove that for any completely monotone function $f$ on a finite lattice, $f^{\alpha }$ is completely monotone for all $\alpha \ge c$, where $c$ is explicitly described. For finite distributive lattices we show that the bound $c$ is sharp. Important examples of completely monotone functions are void functionals of random closed sets. We prove that if $V_{\mathcal{X}}$ is the void functional of a random subset $\mathcal{X}$ of $[n]$, then $V_{\mathcal{X}}^{\alpha }$ is void functional of some random closed set for $\alpha \ge n-1$. The results are analogous to the result of FitzGerald and Horn on Hadamard powers of positive semi-definite matrices. Also, we study the question of approximating an $m$-divisible random set by infinitely divisible random sets, and its generalization to lattices.