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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Fall 2021 • Question 2

Question

Math 6040 • Probability Fall 2021 • Question 2

Let \(Y_{k}\) be i.i.d., \(Y_{k} \ge 0\), and \(\mathbb {E}[Y_{k}] \le 1\) for all \(k\). Prove the existence of the almost sure limit \(M_{\infty } = \lim _{n \to \infty } \prod _{i=1}^{n} Y_{i}\) and describe this random variable explicitly. Do not forget the degenerate cases.

Raw LaTeX

Let \(Y_{k}\) be i.i.d., \(Y_{k} \ge 0\), and \(\mathbb{E}[Y_{k}] \le 1\) for all \(k\). Prove the existence of the almost
sure limit \(M_{\infty} = \lim_{n \to \infty} \prod_{i=1}^{n} Y_{i}\) and describe this random variable explicitly. Do not
forget the degenerate cases.

Topic Tags

Laws of Large Numbers Martingale Convergence Theorems Measure-Theoretic Probability Uniform Integrability and Truncation Methods