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 | References | Similar Articles | Additional Information

Abstract: A probability measure on is called *weakly unimodal* if there exists a constant such that for all ,

(0.1) |

Here, denotes the -ball centered at with radius .

In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of . 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.

**1.**T. W. Anderson,*The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities*, Proc. Amer. Math. Soc.**6**(1955), 170–176. MR**0069229**, 10.1090/S0002-9939-1955-0069229-1**2.**Jean Bertoin,*Lévy processes*, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR**1406564****3.**Marek Kanter,*Unimodality and dominance for symmetric random vectors*, Trans. Amer. Math. Soc.**229**(1977), 65–85. MR**0445580**, 10.1090/S0002-9947-1977-0445580-7**4.**D. KHOSHNEVISAN AND YIMIN XIAO (2002). Level sets of additive Lévy processes,*Ann. Probab.***30**, 62-100.**5.**P. Medgyessy,*On a new class of unimodal infinitely divisible distribution functions and related topics*, Studia Sci. Math. Hungar**2**(1967), 441–446. MR**0222929****6.**Ken-iti Sato,*Class 𝐿 of multivariate distributions and its subclasses*, J. Multivariate Anal.**10**(1980), no. 2, 207–232. MR**575925**, 10.1016/0047-259X(80)90014-7**7.**K.-I. SATO (1999).*Lévy Processes and Infinitely Divisible Distributions,*Cambridge University Press.**8.**Stephen James Wolfe,*On the unimodality of infinitely divisible distribution functions*, Z. Wahrsch. Verw. Gebiete**45**(1978), no. 4, 329–335. MR**511778**, 10.1007/BF00537541**9.**Stephen James Wolfe,*On the unimodality of multivariate symmetric distribution functions of class 𝐿*, J. Multivariate Anal.**8**(1978), no. 1, 141–145. MR**0482940****10.**Stephen James Wolfe,*On the unimodality of infinitely divisible distribution functions. II*, Analytical methods in probability theory (Oberwolfach, 1980) Lecture Notes in Math., vol. 861, Springer, Berlin-New York, 1981, pp. 178–183. MR**655272****11.**Makoto Yamazato,*Unimodality of infinitely divisible distribution functions of class 𝐿*, Ann. Probab.**6**(1978), no. 4, 523–531. MR**0482941**

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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

Email:
davar@math.utah.edu

**Yimin Xiao**

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

Email:
xiao@stt.msu.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06778-3

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