## The Annals of Probability

- Ann. Probab.
- Volume 29, Number 1 (2001), 558-575.

### Inversion de Laplace effective

André Stef and Gérald Tenenbaum

#### Abstract

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

we establish the bound

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.

#### Article information

**Source**

Ann. Probab., Volume 29, Number 1 (2001), 558-575.

**Dates**

First available in Project Euclid: 21 December 2001

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1008956344

**Digital Object Identifier**

doi:10.1214/aop/1008956344

**Mathematical Reviews number (MathSciNet)**

MR1825164

**Zentralblatt MATH identifier**

1020.60008

**Subjects**

Primary: 60E10: Characteristic functions; other transforms

Secondary: 11N25: Distribution of integers with specified multiplicative constraints 40E05: Tauberian theorems, general 44A10: Laplace transform

**Keywords**

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

#### Citation

Stef, André; Tenenbaum, Gérald. Inversion de Laplace effective. Ann. Probab. 29 (2001), no. 1, 558--575. doi:10.1214/aop/1008956344. https://projecteuclid.org/euclid.aop/1008956344