Convergence to the brownian motion

Abstract

[en] The Brownian motion is a stochastic process that models the motion of particles suspended in a liquid or a gas. In mathematics, it also plays a vital role in stochastic calculus. This thesis consists in the proving of three different results of convergence towards the Brownian motion. The first one is proving the Donsker’s theorem, for which different notions of convergence, such as weakly convergence or convergence in distribution, are introduced. The second result consists in the proving of a certain type of stochastic processes converging in distribution towards the Brownian motion. For the last result, uniform transport processes are presented and then it is showed that they converge almost surely to the Brownian motion. In addition, a couple of results that extend this almost sure convergence are mentioned.