Upper and Lower Bounds in Exponential Tauberian Theorems

@article{Voss2009UpperAL,
  title={Upper and Lower Bounds in Exponential Tauberian Theorems},
  author={Jochen Voss},
  journal={arXiv: Probability},
  year={2009},
  pages={41-50}
}
  • Jochen Voss
  • Published 2009
  • Mathematics
  • arXiv: Probability
  • 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… CONTINUE READING

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