Poisson processes and a log-concave Bernstein theorem

TL;DR

This paper explores the relationship between log-concave functions and sequences, proving a Bernstein-type theorem that characterizes log-concave measures via Taylor coefficients, and introduces new concavity inequalities inspired by classical theorems.

Contribution

It introduces a novel Bernstein-type theorem linking log-concavity of measures to Taylor coefficients and develops new concavity inequalities for sequences.

Findings

Characterization of Laplace transforms of log-concave measures

New concavity inequalities for sequences

A stochastic variational formula for the Poisson average

Abstract

We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the Pr\'ekopa-Leindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.