Note on Martingale Theory and Applications
Xiandong Zou
TL;DR
This paper explores fundamental properties of martingales, including measure-theoretic aspects and key theorems, and applies these concepts to analyze extinction probabilities in Galton--Watson branching processes.
Contribution
It provides a measure-theoretic perspective on martingales and demonstrates their application to branching process extinction analysis.
Findings
Martingale convergence under measure-theoretic conditions
Application of martingale theory to Galton--Watson processes
Insights into extinction behavior in branching processes
Abstract
This note investigates core properties of martingales, emphasizing the measure-theoretic formulation of conditional expectation, the martingale transform, and the upcrossing lemma. These results lead to the Martingale Convergence Theorem, which we then apply to study the extinction behavior in Galton--Watson branching processes.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGenerative Adversarial Networks and Image Synthesis · Adversarial Robustness in Machine Learning · Machine Learning and Data Classification