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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 10

Question

Math 6010 • Linear Models Fall 2017 • Question 10

Let \(M_n\) be a martingale such that \(\sup _n\mathbb {E}[M_n^2]<\infty \). Prove that there exists a random variable \(M_{\infty }\in L^2\) such that \(M_n\to M_{\infty }\) in \(L^2\). Hint: \(M_n=M_0+\sum _{j=1}^{n}(M_j-M_{j-1})\).

Raw LaTeX

Let \(M_n\) be a martingale such that \(\sup_n\E[M_n^2]<\infty\).
Prove that there exists a random variable \(M_{\infty}\in L^2\) such that
\(M_n\to M_{\infty}\) in \(L^2\). Hint:
\(M_n=M_0+\sum_{j=1}^{n}(M_j-M_{j-1})\).

Topic Tags

Exponential Martingales L2 Convergence and Covariance Bounds Martingales and Supermartingales Uniform Integrability and Truncation Methods