Electronic Communications in Probability

Uniform Upper Bound for a Stable Measure of a Small Ball

Michal Ryznar and Tomasz Zak

Full-text: Open access

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

Permanent link to this document
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
Primary: 60B11: Probability theory on linear topological spaces [See also 28C20]
Secondary: 69E07

Keywords
stable measure small ball

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Ryznar, Michal; Zak, Tomasz. Uniform Upper Bound for a Stable Measure of a Small Ball. Electron. Commun. Probab. 3 (1998), paper no. 9, 75--78. doi:10.1214/ECP.v3-995. https://projecteuclid.org/euclid.ecp/1456935915


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References

  • P. Hitczenko, S. Kwapien, W.N. Li, G. Schechtman, T. Schlumprecht and J. Zinn (1998), Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables. Electronic Journal of Probability 3, 1-26, Paper 2.
  • R. LePage, M. Woodroofe and J. Zinn (1981), Convergence to a stable distribution via order statistics. Ann. Probab.9,624-632.
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  • N. Cressie (1975), A note on the behaviour of the stable distribution for small index $alpha$. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33,61-64.