In this expository note we give a short, self-contained, and elementary proof of the strong law of large numbers under a power law decay hypothesis for joint second moments. The result is related to the classical one by Lyons. However, we also provide a rate of convergence. Our proof does not use maximal inequalities and is instead inspired by the method of multiscale large versus small field decompositions in constructive quantum field theory. As a hopefully entertaining application, we also include a short derivation of the so-called « Infinite Monkey Theorem ».
Received: Juin 26, 2018.
Published online: October 20, 2018.
Abdelmalek AbdesselamDepartment of Mathematics, P. O. Box 400137, University of Virginia, Charlottesville, VA 22904-4137, USA.