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

TL;DR

This paper proves that a class of functions related to Bernstein functions are complete and extends a construction of nested regenerative sets associated with subordinators, providing new insights into their structure and properties.

Contribution

The paper provides simple proofs that certain functions are complete Bernstein functions and generalizes Marchal's nested regenerative set construction to all complete Bernstein functions.

Findings

Proved that lphare complete Bernstein functions.

Extended the construction of nested regenerative sets to all complete Bernstein functions.

Provided simplified proofs for properties of these functions.

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.