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Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes

Authors:

Institution:

Department of Mathematics, 155 S. 1400 E., JWB 233, University of Utah, Salt Lake City, Utah 84112-0090 ; Department of Statistics and Probability, A--413 Wells Hall, Michigan State University, East Lansing, Michigan 48824

Abstract:

A probability measure $\mu$ on $\mathbb{R}^d$ is called weakly unimodal if there exists a constant $\kappa \ge 1$ such that for all

$\begin{displaymath}\sup_{a\in\mathbb{R}^d} \mu(B(a, r)) \le \kappa \mu(B(0, r)). \end{displaymath}$

(0.1)

Here,

denotes the $\ell^\infty$ -ball centered at $a\in\mathbb{R}^d$ with radius

In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of $\mathbb{R}^d$ . In particular, we use this to prove that every symmetric infinitely divisible distribution is weakly unimodal. This result is then applied to improve some recent results of the authors on capacities and level sets of additive Lévy processes.

Keywords:

Weak unimodality infinitely divisible distributions additive L\'evy processes potential theory

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