El movimiento browniano como límite del paseo aleatorio: el teorema de Donsker

Abstract

[en] Brownian motion is a continuous time stochastic process with no memory, that is, the current state of the process is not influenced by its past. This property is named ”Markov property”. The main purpose of this paper is to obtain a Brownian motion from a discrete time stochastic process named Random Walk. The Random Walk is also a Markov process. To achieve this goal, we are going to study weak convergence on metric spaces and, in particular, on $C$ ([0, 1]). Brownian motion is the obtained as a weak limit of a sequence of linear interpolations of Random Walk normalized in a suitable way. This is Donsker’s theorem (1951).