Weak approximations towards Gaussian processes

Abstract

Weak convergence, or convergence in distribution, is the type of convergence in which, unlike the other well-known types of convergence, whether a sequence of random variables convergences in distribution or not depends only on their distributions. In addition to its intrinsic mathematical interest, weak convergence is precisely the kind of convergence that we encounter in the renowned central limit theorem. This thesis consists in profoundly studying the theory of weak convergence of probability measures and then use all this information to prove three different results of weak approximations towards different types of Gaussian processes. The first result is proving that we can weakly approximate a complex Brownian motion using processes with independent increments, and then use this result by applying it to the case of L´evy processes. The second one consists of the proving of a certain type of stochastic processes, called empirical processes, weakly converging to a Gaussian process with given properties. For the last result, we will prove that another type of stochastic processes, called renewal processes, weakly converge to the Brownian motion.