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.

References in corpus (4)

• Coupling property and gradient estimates of Lévy processes via the symbol
• Strong convergence of the Euler--Maruyama approximation for a class of Lévy-driven SDEs
• On the domain of fractional Laplacians and related generators of Feller processes
• On the integral modulus of infinitely divisible distributions

Cited by in corpus (4)

• Strong convergence of the Euler--Maruyama approximation for a class of Lévy-driven SDEs
• Schauder estimates for equations associated with Lévy generators
• Schauder estimates for Poisson equations associated with non-local Feller generators
• A Liouville theorem for Lévy generators