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Measure concentration for stable laws with index close to 2. (English) Zbl 1060.60011

Summary: We give upper bounds for the probability \(\mathbb P(|f(X)-Ef(X)|>x)\), where \(X\) is a stable random variable with index close to 2 and \(f\) is a Lipschitz function. While the optimal upper bound is known to be of order \(1/x^\alpha\) for large \(x\), we establish, for smaller \(x\), an upper bound of order \(\exp(-x^\alpha/2)\), which relates the result to the Gaussian concentration.

MSC:

60E07 Infinitely divisible distributions; stable distributions
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