Upper and Lower Bounds in Exponential Tauberian Theorems

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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^-λX) ~exp(rλ^α) for λ→∞ and P(X≤ε) ~ exp(s/ε^β) for ε↓0 are in some sense equivalent (for 1/α= 1/β+ 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.

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