Effective estimation of some oscillatory integrals related to infinitely divisible distributions

We present a practical framework to prove, in a simple way, two-term asymptotic expansions for Fourier integrals I ( t ) = ∫ R ( e i t ϕ ( x ) - 1 ) d μ ( x ) , where μ is a probability measure on R and ϕ 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 fraction coefficients.