# Sebastian Franzén: A comparison of two proofs of Donsker’s theorem

BSc thesis presentation

**Time: **
Tue 2020-06-02 13.30 - 14.30

**Lecturer: **
Sebastian Franzén

**Location: **
Zoom, meeting ID: 618 8346 8694

**Supervisor: **
Daniel Ahlberg

### Abstract

This thesis explores Donsker's theorem: a theorem in the subject of stochastic processes that relates a Brownian motion to a limit of random walks. It states that a random walk, appropriately rescalled in time and space, and linearly interpolated between its values at integer times, converges weakly to a Brownian motion.

There are two quite different approaches to proving the theorem that involve entirely different techniques. Both of them will be described and some of the theory involved will be presented. As will be shown in this paper, one approach will prove to be a possible construction of Brownian motion. The second approach assumes its prior existence, but instead it provides, as a corollary, the Central limit theorem.