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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2009 • Question 7

Question

Math 6010 • Linear Models Fall 2009 • Question 7

Let \(X_1,X_2,\ldots \) be i.i.d. uniform\((0,2)\), and put \(M_n:=X_1X_2\cdots X_n\) for each \(n\geq 1\). Show that \(\{M_n\}_{n=1}^{\infty }\) is a martingale. What does the martingale convergence theorem tell us about \(\lim _{n\to \infty }M_n\)? (In particular, evaluate this limit.)

Raw LaTeX

Let \(X_1,X_2,\ldots\) be i.i.d. uniform\((0,2)\), and put
\(M_n:=X_1X_2\cdots X_n\) for each \(n\geq1\). Show that \(\{M_n\}_{n=1}^{\infty}\)
is a martingale. What does the martingale convergence theorem tell us about
\(\lim_{n\to\infty}M_n\)? (In particular, evaluate this limit.)

Topic Tags

Martingale Convergence Theorems Martingales and Supermartingales Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits