The speed of convergence in the renewal theorem

TL;DR

This paper investigates the convergence speed in Kesten's renewal theorem for probability measures on the real line, focusing on diophantine properties and Fourier-Laplace transforms to understand the rate of convergence.

Contribution

It establishes a connection between diophantine properties of measures and the convergence speed in renewal theorems, providing new insights into the spectral analysis involved.

Findings

Identifies the role of Fourier-Laplace transform points near the imaginary axis.

Links diophantine properties to convergence rates.

Applies findings to Kesten's renewal theorem on R.

Abstract

In this article we study a diophantine property of probability measures on R. We will always assume that the considered measures have an exponential moment and a drift. We link this property to the points in C close to the imaginary axis where the Fourier-Laplace transform of those measures take the value 1 and finally, we apply this to the study of the speed in Kesten's renewal theorem on R.