A probabilistic proof of Schoenberg's theorem

Abstract

Assume that $g(|\xi {|}^2)$, $\xi \in {\mathbb{R}}^k$, is for every dimension $k\in \mathbb{N}$ the characteristic function of an infinitely divisible random variable $X^k$. By a classical result of Schoenberg $f:=-\log g$ is a Bernstein function. We give a simple probabilistic proof of this result starting from the observation that $X^k=X_1^k$ can be embedded into a Lévy process $(X_t^k{)}_{t\ge 0}$ and that Schoenberg's theorem says that $(X_t^k{)}_{t\ge 0}$ is subordinate to a Brownian motion. A key ingredient of our proof are concrete formulae which connect the transition densities, resp., Lévy measures of subordinated Brownian motions across different dimensions. As a by-product of our proof we obtain a gradient estimate for the transition semigroup of a subordinated Brownian motion.