Note on Martingale Theory and Applications

Xiandong Zou
2/5/2026
cs.LGcs.AImath.PR

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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.

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Cite this paper

@article{zou2026note,
  title  = {Note on Martingale Theory and Applications},
  author = {Xiandong Zou},
  year   = {2026},
  eprint = {2602.05774},
  archivePrefix = {arXiv},
  url    = {http://arxiv.org/abs/2602.05774v3}
}

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