Bachelor of Science

There are several objectives to this project. The first is to investigate the properties of random walks and Brownian motion and to express Brownian motion as a limit of random walks. Secondly, we make use of the Borel-Cantelli lemmas to obtain an asymptotic boundary (a rather crude one) on the random walk and Brownian motion. Lastly, the Poisson clumping heuristic. together with the Ornstein-Uhlenbeck process, is used to derive a more refined asymptotic boundary for the Brownian motion. It must be stressed at this point that the approach used in this work is largely of a heuristic nature. The reader should not expect to see lengthy and rigorous proofs. Instead, appropriate approximation and estimations are used to deduce the desired results.