Non-asymptotic control of the cumulative distribution function of Lévy processes

Abstract

We propose non-asymptotic controls of the cumulative distribution function $P(|X_t|\ge \epsilon )$, for any $t>0$, $\epsilon >0$ and any Lévy process $X$ such that its Lévy density is bounded from above by the density of an $\alpha$-stable type Lévy process in a neighborhood of the origin. The results presented are non-asymptotic and optimal, they apply to a large class of Lévy processes.