Inversion de Laplace effective

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 $$\sup_{0 \le u \le \kappa}|\widehat{F}(u) - \widehat{G} (u) | \le \varepsilon, \qquad \widehat{F}(u) + \widehat{G} (u) \le h(u) \qquad (-L \le u \le L),$$ 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.