# Cherchez la martingale

Stuff about probability and orthogonality

November 26, 2019 — December 1, 2019

Like Markov processes, a weirdly useful class of stochastic processes. Often you can find a martingale within some stochastic process, or construct a martingale from a stochastic process and prove something nifty thereby; this idea connects and solves a bunch of tricky problems at once.

TODO: examples, maybe a CLT and something else wacky like the life table estimators of (Aalen 1978).

I am indebted to Saif Syed for setting my head straight about the utility of martingales, and Kevin Ross who, in part of Amir Dembo’s course materials, was the one whose explanation of the orthogonality interpretation of martingales finally communicated the neatness of this idea to me.

TBC.

## 1 Local martingales

The classical gambling strategy (double-down until you win) is in fact a *local* martingale.

## 2 References

*The Annals of Statistics*.

*Annals of the Institute of Statistical Mathematics*.

*Measure theory and probability theory*.

*Bernoulli*.

*Fundamentals of the Theory of Structured Dependence Between Stochastic Processes*. Encyclopedia of Mathematics and Its Applications.

*Le Calcul Des Probabilités Et Ses Applications*. Colloques Internationaux Du Centre National de La Recherche Scientifique, No. 13.

*Stochastic Processes and Their Applications*.

*Stochastic Processes and Their Applications*.

*Selected Works of C.C. Heyde*. Selected Works in Probability and Statistics.

*arXiv:1604.08305 [Math]*.

*Séminaire de Probabilités XXXI*. Lecture Notes in Mathematics 1655.

*The Annals of Probability*.

*Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation*. Grundlehren Der Mathematischen Wissenschaften : A Series of Comprehensive Studies in Mathematics 345.

*arXiv:1602.00721 [Cs, Math]*.

*arXiv:1803.05646 [Math]*.

*The Annals of Probability*.

*arXiv:1202.3223 [Math]*.

*Physica A: Statistical and Theoretical Physics*.

*Statistica Neerlandica*.

*Foundations and Trends in Communications and Information Theory*.

*Probability Theory and Related Fields*.

*Optimizing Methods in Statistics*.

*The Econometrics Journal*.

*Quantitative Finance*.

*The Annals of Statistics*.