Electronic Communications in Probability

Electronic Communications in Probability > Vol. 3 (1998) > Paper 9

Uniform Upper Bound for a Stable Measure of a Small Ball

Michal Ryznar, Wroclaw University of Technology
Tomasz Zak, Wroclaw University of Technology

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 $mu(tB)le R(b)tmu(B)$ for all $0 < t < 1$. We prove that the above inequality holds but the constant $R$ must depend also on $alpha$.

Full text: PDF

Pages: 75-78

Published on: September 16, 1998

Bibliography

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. Math. Review 82k:60049
M. Lewandowski, M. Ryznar and T. Zak (1992), Stable measure of a small ball. Proc. Amer. Math. Soc.115,489-494. Math. Review 92i:60004
N. Cressie (1975), A note on the behaviour of the stable distribution for small index $alpha$. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33,61-64. Math. Review 52:1825

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Electronic Communications in Probability. ISSN: 1083-589X