A probabilistic proof of Schoenberg's theorem

Abstract

Assume that g(|ξ|2), ξ∈Rk, is for every dimension k∈N the characteristic function of an infinitely divisible random variable Xk. 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 Xk=X1k can be embedded into a Lévy process (Xtk)t≥0 and that Schoenberg's theorem says that (Xtk)t≥0 is subordinate to a Brownian motion. Key ingredients in 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.