A probabilistic proof of Schoenberg's theorem.

Assume that g (| ξ | 2) , ξ ∈ R k , is for every dimension k ∈ 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 ≥ 0 and that Schoenberg's theorem says that (X t k) 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. [ABSTRACT FROM AUTHOR]