A Tutorial on Brownian Motion for Biostatisticians

TL;DR

This tutorial offers a comprehensive overview of Brownian Motion, covering foundational concepts, advanced topics, and key theorems relevant for biostatisticians working with stochastic processes.

Contribution

It provides an in-depth, rigorous tutorial on Brownian Motion tailored for biostatisticians, including advanced topics and proofs not commonly found in standard texts.

Findings

Detailed explanations of Brownian motion properties

In-depth coverage of advanced topics like Karhunen-Loeve expansion

Discussion of key theorems and their implications

Abstract

This manuscript provides an in-depth exploration of Brownian Motion, a fundamental stochastic process in probability theory for Biostatisticians. It begins with foundational definitions and properties, including the construction of Brownian motion and its Markovian characteristics. The document delves into advanced topics such as the Karhunen-Loeve expansion, reflection principles, and Levy's modulus of continuity. Through rigorous proofs and theorems, the manuscript examines the non-differentiability of Brownian paths, the behavior of zero sets, and the significance of local time. The notes also cover important results like Donsker's theorem and Blumenthal's 0-1 law, emphasizing their implications in the study of stochastic processes.