Upper and Lower Bounds in Exponential Tauberian Theorems

TL;DR

This paper explores the relationship between the asymptotic behavior of Laplace transforms and distribution tails for positive random variables, extending De Bruijn's Tauberian theorem to more general cases and illustrating applications in large deviations.

Contribution

It generalizes De Bruijn's exponential Tauberian theorem to cases with different upper and lower limits, providing new bounds and applications.

Findings

Extended Tauberian bounds for non-matching limits

Derived large deviation estimates using the generalized results

Illustrated applications in probabilistic tail analysis

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 $E(e^{-\lambda X}) \sim \exp(r\lambda^\alpha)$ for $\lambda\to\infty$ and $P(X\leq\epsilon) \sim \exp(s/\epsilon^\beta)$ for $\epsilon\downarrow0$ are in some sense equivalent (for $1/\alpha = 1/\beta + 1$) and gives a relation between the constants $r$ and $s$. We illustrate how this result can be used to obtain simple 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.

Equations

$$r=\underset{\lambda \to \infty }{\lim }\frac{1}{{\lambda }^{\alpha }}\log E({\mathrm{e}}^{-\lambda X}\cdot 1_A)\le 0(1):tauber-laplace=E01$$

Full text

1 Introduction

Tauberian theorems (see for example the monograph Ko04) describe the connection between the behaviour of a positive random variable near zero and the behaviour of its Laplace transform near infinity. From De Bruijn’s Tauberian theorem (theorem 4.12.9 in Bingham-Goldie-Teugels87) we can easily conclude the following result.

Theorem 1

Let $X\geq 0$ be a random variable on a probability space $(\Omega,\mathcal{A},P)$, $A\in\mathcal{A}$ an event with $P(A)>0$ and $\alpha\in(0,1)$, $\beta>0$ with $\frac{1}{\alpha}=\frac{1}{\beta}+1$. Then the limit

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