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February 2001 Inversion de Laplace effective
André Stef, Gérald Tenenbaum
Ann. Probab. 29(1): 558-575 (February 2001). DOI: 10.1214/aop/1008956344


Let $F,G$ be arbitrary distribution functions on the real line and let $\widehat{F},\widehat{G}$ denote their respective bilateral Laplace transforms. Let $\kappa > 0$ and let $h : \mathbb{R}^+ \to \mathbb{R}^+$ be continuous, non-decreasing, and such that $h(u) \ge Au^4$ for some $A > 0$ and all $u \ge 0$. Under the assumptions that $$\sup_{0 \le u \le \kappa}|\widehat{F}(u) - \widehat{G} (u) | \le \varepsilon, \qquad \widehat{F}(u) + \widehat{G} (u) \le h(u) \qquad (-L \le u \le L),$$ we establish the bound $$\sup_{u \in \mathbb{R}}|\widehat{F}(u) - \widehat{G} (u) | \le CQ_G(l)$$ where $C$ is a constant depending at most on $\kappa$ and $A$, $Q_G$ is the concentration function of $G$, and $l := (\log L) /L + (\log W) /W$, with $W$ any solution to $h(W) = 1/\epsilon$. Improving and generalizing an estimate of Alladi, this result provides a Laplace transform analogue to the Berry-Esseen inequality, related to Fourier transforms. The dependence in $\epsilon$ is optimal up to the logarithmic factor log $W$. A number-theoretic application, developed in detail elsewhere, is described. It concerns so-called lexicographic integers, whose characterizing property is that their divisors are ranked according to size and valuation of the largest prime factor. The above inequality furnishes, among other informations, an effective Erdös-Kac theorem for lexicographical integers.


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André Stef. Gérald Tenenbaum. "Inversion de Laplace effective." Ann. Probab. 29 (1) 558 - 575, February 2001.


Published: February 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1020.60008
MathSciNet: MR1825164
Digital Object Identifier: 10.1214/aop/1008956344

Primary: 60E10
Secondary: 11N25 , 40E05 , 44A10

Keywords: Berry-Esseen inequality , concentration functions , effective inversion , Laplace transform , lexicographical integers , one-sided L1-approximation

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 1 • February 2001
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