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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Spring 2022 • Question 3

Question

Math 6040 • Probability Spring 2022 • Question 3

Consider a probability space \((\Omega , \mathcal {F}, P)\). Let \(X \in L^{1}\). Suppose \((\mathcal {F}_{n})_{n \in \mathbb {N}}\) is an increasing sequence of \(\sigma \)-algebras such that \(\cup _{n \in \mathbb {N}}\mathcal {F}_{n}\) generates \(\mathcal {F}\). Prove the Doob martingale convergence theorem that states that \(\mathbb {E}[X \mid \mathcal {F}_{n}]\) converges almost surely to \(X\).

Raw LaTeX

Consider a probability space \((\Omega, \mathcal{F}, P)\). Let \(X \in L^{1}\). Suppose \((\mathcal{F}_{n})_{n \in \mathbb{N}}\) is an increasing
sequence of \(\sigma\)-algebras such that \(\cup_{n \in \mathbb{N}}\mathcal{F}_{n}\) generates \(\mathcal{F}\). Prove the Doob martingale
convergence theorem that states that \(\mathbb{E}[X \mid \mathcal{F}_{n}]\) converges almost surely to \(X\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Martingale Convergence Theorems Martingales and Supermartingales Measure-Theoretic Probability