## Electronic Communications in Probability

### Uniform Upper Bound for a Stable Measure of a Small Ball

#### Abstract

P. Hitczenko, S.Kwapien, W.N.Li, G.Schechtman, T.Schlumprecht and J.Zinn stated the following conjecture. Let $\mu$ be a symmetric $\alpha$-stable measure on a separable Banach space and $B$ a centered ball such that $\mu(B)\le b$. Then there exists a constant $R(b)$, depending only on $b$, such that $\mu(tB)\le R(b)t\mu(B)$ for all $0 \lt t \lt 1$. We prove that the above inequality holds but the constant $R$ must depend also on $\alpha$.

#### Article information

Source
Electron. Commun. Probab., Volume 3 (1998), paper no. 9, 75-78.

Dates
Accepted: 16 September 1998
First available in Project Euclid: 2 March 2016

https://projecteuclid.org/euclid.ecp/1456935915

Digital Object Identifier
doi:10.1214/ECP.v3-995

Mathematical Reviews number (MathSciNet)
MR1645592

Zentralblatt MATH identifier
0907.60009

Subjects
Secondary: 69E07

Keywords
stable measure small ball

Rights
• N. Cressie (1975), A note on the behaviour of the stable distribution for small index $alpha$. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33,61-64.