We give upper bounds for the probability $P(|f(X)-Ef(X)| > x)$, 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.
"Measure Concentration for Stable Laws with Index Close to 2." Electron. Commun. Probab. 10 29 - 35, 2005. https://doi.org/10.1214/ECP.v10-1129