Brownian Motion Construction

This lecture covers the construction of Brownian motion as a standard Gaussian process, characterized by a continuous sample path and specific covariance properties. The process is built using sequences of independent standard normal random variables, ensuring convergence and continuity. Various lemmas are introduced to demonstrate the properties of the constructed process, leading to the establishment of a Gaussian process with desired covariance. The lecture emphasizes the convergence of the constructed process and its uniformity over compact intervals, highlighting the key characteristics of Brownian motion.