# Upper and Lower Bounds in Exponential Tauberian Theorems - Mathematics > Probability

Abstract: In this text we study, for positive random variables, the relation betweenthe behaviour of the Laplace transform near infinity and the distribution nearzero. A result of De Bruijn shows that $Ee^{-\lambda X} \sim\expr\lambda^\alpha$ for $\lambda\to\infty$ and $PX\leq\epsilon \sim\exps-\epsilon^\beta$ for $\epsilon\downarrow0$ are in some sense equivalentfor $1-\alpha = 1-\beta + 1$ and gives a relation between the constants $r$and $s$. We illustrate how this result can be used to obtain simple largedeviation results. For use in more complex situations we also give ageneralisation of De Bruijn-s result to the case when the upper and lowerlimits are different from each other.

Author: Jochen Voss

Source: https://arxiv.org/