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 $${\varphi }^{(\alpha )}(\lambda ):=\exp \left[{\int }_0^1\frac{\lambda -1}{1+(\lambda -1)x}\alpha (x)dx\right],\lambda >0$$ 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 ${\mathcal{R}}^{(\alpha )}\stackrel{\text{law}}{=}\overline{\{S_t^{(\alpha )}:t\ge 0\}}$ ($S^{(\alpha )}$ is the subordinator with Laplace exponent ${\varphi }^{(\alpha )}$) and ${\mathcal{R}}^{(\alpha )}\subset {\mathcal{R}}^{(\beta )}$ whenever $\alpha \le \beta$. We give two simple proofs showing that ${\varphi }^{(\alpha )}$ is a complete Bernstein function and extend Marchal's construction to all complete Bernstein functions.

to appear in Electron. Comm. Probab

References in corpus (1)

• On exponential functionals, harmonic potential measures and undershoots of subordinators