Sanz-Solé, MartaPitarch Ferreiro, Marta2019-09-202019-09-202019-01-17https://hdl.handle.net/2445/140617Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Marta Sanz[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).52 p.application/pdfspacc-by-nc-nd (c) Marta Pitarch Ferreiro, 2019http://creativecommons.org/licenses/by-nc-nd/3.0/es/Moviment browniàTreballs de fi de grauTeorema del límit centralRutes aleatòries (Matemàtica)Brownian movementsBachelor's thesesCentral limit theoremRandom walks (Mathematics)info:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess