Effective estimation of some oscillatory integrals related to infinitely divisible distributions

TL;DR

This paper introduces a practical method for deriving two-term asymptotic expansions of certain Fourier integrals linked to infinitely divisible distributions, with applications to limit laws in number theory.

Contribution

It provides a simple framework for asymptotic analysis of Fourier integrals associated with probability measures, extending to applications in limit laws for continued fractions.

Findings

Established a straightforward approach for asymptotic expansions

Applied framework to limit laws in rational continued fractions

Connected Fourier integral analysis with Levy's continuity theorem

Abstract

We present a practical framework to prove, in a simple way, two-terms asymptotic expansions for Fourier integrals $$ {\mathcal I}(t) = \int_{\mathbb R}({\rm e}^{it\phi(x)}-1) {\rm d} \mu(x) $$ where $\mu$ is a probability measure on $\mathbb{R}$ and $\phi$ is measurable. This applies to many basic cases, in link with Levy's continuity theorem. We present applications to limit laws related to rational continued fractions coefficients.