A Tutorial on Brownian Motion for Biostatisticians
Elvis Han Cui
TL;DR
This paper provides a rigorous tutorial on Brownian motion $B(t)$, detailing its formal definition, construction, and core properties, and highlighting its relevance to biostatistics. It develops methodology by examining existence/construction, Markov and strong Markov properties, and the Karhunen-Loève expansion, along with pathwise results such as the reflection principle and local time. Key results include Blumenthal's 0-1 law and Donsker's theorem, as well as explicit examples for zero sets and hitting times. Together, these findings provide a practical stochastic toolkit for biostatisticians, enabling diffusion modeling, empirical process reasoning, and stochastic calculus in real-world bioscience scenarios.
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.