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

Abstract

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