A probabilistic proof of Schoenberg's theorem

TL;DR

This paper provides a simple probabilistic proof of Schoenberg's theorem, showing that certain characteristic functions are linked to Bernstein functions via subordinated Brownian motions, and derives a gradient estimate for their transition semigroups.

Contribution

It offers a novel probabilistic approach to Schoenberg's theorem using Le9vy process embeddings and explicit formulas connecting transition densities across dimensions.

Findings

Probabilistic proof of Schoenberg's theorem.

Explicit formulas for transition densities and Le9vy measures.

Gradient estimate for subordinated Brownian motion's transition semigroup.

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 L\'evy process $(X_t^k)_{t\geq 0}$ and that Schoenberg's theorem says that $(X_t^k)_{t\geq 0}$ is subordinate to a Brownian motion. A key ingredient of our proof are concrete formulae which connect the transition densities, resp., L\'evy 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.