Effective estimation of some oscillatory integrals related to infinitely divisible distributions

Abstract

We present a practical framework to prove, in a simple way, two-terms asymptotic expansions for Fourier integrals $$\mathcal{I}(t)={\int }_{\mathbb{R}}({\mathrm{e}}^{it\varphi (x)}-1)\mathrm{d}\mu (x)$$ where $\mu$ is a probability measure on $\mathbb{R}$ and $\varphi$ 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.

9 pages. arXiv admin note: text overlap with arXiv:1903.00457