Complete Bernstein functions and subordinators with nested ranges. A note on a paper by P. Marchal

Abstract

Let $α:[0,1]\to [0,1]$ be a measurable function. It was proved by P. Marchal \cite{Mar15} that the function $$ ϕ^{(α)}(λ):=\exp\left[ \int_0^1\frac{λ-1}{1+(λ-1)x}\,α(x)\,d x \right],\quad λ>0 $$ is a special Bernstein function. Marchal used this to construct, on a single probability space, a family of regenerative sets $\mathcal R^{(α)}$ such that $\mathcal{R}^{(α)} \stackrel{\text{law}}{=} \overline{\{S^{(α)}_t:t\geq 0\}}$ ($S^{(α)}$ is the subordinator with Laplace exponent $ϕ^{(α)}$) and $\mathcal R^{(α)}\subset \mathcal R^{(β)}$ whenever $α\leqβ$. We give two simple proofs showing that $ϕ^{(α)}$ is a complete Bernstein function and extend Marchal's construction to all complete Bernstein functions.