Geodesic
On Finite Range Stable-Type Concentration
Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 4, pp. 711-735 Cet article a éte moissonné depuis la source Math-Net.Ru

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We first study the deviation probability $P\{f(X)-E[f(X)]\ge x\}$, where $f$ is a Lipschitz (for the Euclidean norm) function defined on $R^d$ and $X$ is an $\alpha$-stable random vector of index $\alpha \in (1,2)$. We show that this probability is upper bounded by either $e^{-cx^{\alpha/(\alpha-1)}}$ or $e^{-cx^\alpha}$ according to $x$ taking small values or being in a finite range interval. We generalize these finite range concentration inequalities to $P\{F-m(F)\ge x\}$ where $F$ is a stochastic functional on the Poisson space equipped with a stable Lйvy measure of index $\alpha\in(0,2)$ and where $m(F)$ is a median of $F$.
Keywords: concentration of measure phenomenon, stable random vectors, infinite divisibility. 
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     author = {J.-Ch. Breton and Ch. Houdr\'e},
     title = {On {Finite} {Range} {Stable-Type} {Concentration}},
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     volume = {52},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2007_52_4_a4/}
}
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J.-Ch. Breton; Ch. Houdré. On Finite Range Stable-Type Concentration. Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 4, pp. 711-735. http://geodesic.mathdoc.fr/item/TVP_2007_52_4_a4/

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