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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 10

Question

Math 6010 • Linear Models 2005 • Question 10

Suppose \(\{X_n\}_{n=1}^{\infty }\) is a sequence of random variables that has the property that \(\sup _n|X_n|\le 1\) a.s. Then use Doob’s decomposition to prove that \(\sum _n X_n\) converges a.s. iff \(\sum _n\mathbb {E}[X_n\mid X_1,\ldots ,X_{n-1}]\) converges a.s.

Raw LaTeX

Suppose \(\{X_n\}_{n=1}^{\infty}\) is a sequence of random variables that
has the property that \(\sup_n|X_n|\le 1\) a.s. Then use Doob's decomposition to prove
that \(\sum_n X_n\) converges a.s. iff
\(\sum_n\E[X_n\mid X_1,\ldots,X_{n-1}]\) converges a.s.

Topic Tags

Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Uniform Integrability and Truncation Methods