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.
Categories: math.PR