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2009 Upper and Lower Bounds in Exponential Tauberian Theorems
Jochen Voss
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Tbilisi Math. J. 2: 41-50 (2009). DOI: 10.32513/tbilisi/1528768840

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.

Acknowledgment

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].

Citation

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Jochen Voss. "Upper and Lower Bounds in Exponential Tauberian Theorems." Tbilisi Math. J. 2 41 - 50, 2009. https://doi.org/10.32513/tbilisi/1528768840

Information

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

zbMATH: 1203.60031
MathSciNet: MR2574871
Digital Object Identifier: 10.32513/tbilisi/1528768840

Subjects:
Primary: 60F10
Secondary: 44A10

Rights: Copyright © 2009 Tbilisi Centre for Mathematical Sciences

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