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Martingale

Doob upcrossing inequality

Martingale convergence theorem

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For a decreasing sequence of sigma-algebras $\mathcal F_n$ and an integrable random variable $X$,
$$
\mathbb E[X\mid\mathcal F_n]
\longrightarrow
\mathbb E[X\mid\bigcap_n\mathcal F_n]
$$
almost surely and in $L^1$.


Reverse martingale convergence theorem

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 108 / 4 / Solution

Tail sigma-algebra of a measurable partition

Reverse martingale convergence theorem

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Reverse martingale convergence theorem

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