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

Authors: Davar Khoshnevisan and Yimin Xiao
Journal: Proc. Amer. Math. Soc. 131 (2003), 2611-2616
MSC (2000): Primary 60G60; Secondary 60J45
Published electronically: November 6, 2002
MathSciNet review: 1974662
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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 $r>0$,

\begin{displaymath}\sup_{a\in\mathbb{R}^d} \mu(B(a, r)) \le \kappa \mu(B(0, r)). \end{displaymath} (0.1)

Here, $B(a, r)$ denotes the $\ell^\infty$-ball centered at $a\in\mathbb{R}^d$ with radius $r>0$.

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.

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Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, 155 S. 1400 E., JWB 233, University of Utah, Salt Lake City, Utah 84112-0090

Yimin Xiao
Affiliation: Department of Statistics and Probability, A–413 Wells Hall, Michigan State University, East Lansing, Michigan 48824

Keywords: Weak unimodality, infinitely divisible distributions, additive L\'evy processes, potential theory
Received by editor(s): August 18, 2001
Received by editor(s) in revised form: March 21, 2002
Published electronically: November 6, 2002
Additional Notes: The authors’ research was partially supported by grants from NSF and NATO
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2002 American Mathematical Society