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January, 1986 Asymptotic Behaviour of Stable Measures Near the Origin
M. Ryznar
Ann. Probab. 14(1): 287-298 (January, 1986). DOI: 10.1214/aop/1176992628


We investigate the lower tail of $q_r = (\sum^\infty_{i=1} | \alpha_i\theta_i|^r)^{1/r}$ seminorms on $R^\infty$, where $r \geq 1$ and $\theta_i$ are standard $p$-stable real random variables. We prove that for $p < r \leq 2$ we have $P\{q_r \leq t\} \geq \exp\{-ct^{-pr/(r-p)}\}$ in some neighbourhood of 0, where $c$ is a nonnegative constant. If $r \leq p$, then for any positive, increasing function $f$, we can find $q_r$ such that $P\{q_r \leq t\} \leq f(t)$ for $t \leq 1$. We also give a new characterization of Banach spaces of stable type $p$ in terms of the behaviour of $\mu\{\| \cdot \| \leq t\}$ near 0, where $\mu$ is a symmetric and $p$-stable measure.


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M. Ryznar. "Asymptotic Behaviour of Stable Measures Near the Origin." Ann. Probab. 14 (1) 287 - 298, January, 1986.


Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0591.60007
MathSciNet: MR815971
Digital Object Identifier: 10.1214/aop/1176992628

Primary: 60B11
Secondary: 60E07

Keywords: lower tail , seminorm , space of stable type $p$ , Stable measures

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 1 • January, 1986
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