Matilda Colarieti Tosti: Convergence of random series

Abstract: This thesis studies the convergence of random series, where the terms are given by random variables rather than fixed numbers. The main fokus is the development and application of martingale methods, particularly in L^2, to reduce convergence questions to tractable conditions such as boundedness of the underlying random variances. After reviewing foundational concepts from real analysis and probability, we present key martingale properties and demonstrate how they yield general criteria for almost sure convergence of random series.