## Tbilisi Mathematical Journal

### Upper and Lower Bounds in Exponential Tauberian Theorems

Jochen Voss

#### Abstract

In this text we study, for positive random variables, the relation between the behaviour of the Laplace transform near infinity and the distribution near zero. A result of De Bruijn shows that $\mathrm{E}(\mathrm{e}^{-\lambda X}) \sim \exp(r\lambda^\alpha)$ for $\lambda\to\infty$ and ${P}(X\leq\varepsilon) \sim \exp(s/\varepsilon^\beta)$ for $\varepsilon\downarrow0$ are in some sense equivalent (for $1/\alpha = 1/\beta + 1$) and and gives a relation between the constants $r$ and $s$. We illustrate how this result can be used to obtain simple large deviation results. For use in more complex situations we also give a generalisation of De Bruijn's result to the case when the upper and lower limits are different from each other.

#### Note

I want to thank the anonymous referees for pointing out that my original proof for the case $\alpha=1/2$ and $\beta=1$ could be changed to give the more general result presented here, and also for pointing me to the references [5] and [1].

#### Article information

Source
Tbilisi Math. J., Volume 2 (2009), 41-50.

Dates
Received: 20 April 2009
Revised: 30 September 2009
Accepted: 26 October 2009
First available in Project Euclid: 12 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1528768840

Mathematical Reviews number (MathSciNet)
MR2574871

Zentralblatt MATH identifier
1203.60031

Subjects
Primary: 60F10: Large deviations
Secondary: 44A10: Laplace transform

#### Citation

Voss, Jochen. Upper and Lower Bounds in Exponential Tauberian Theorems. Tbilisi Math. J. 2 (2009), 41--50. https://projecteuclid.org/euclid.tbilisi/1528768840