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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Spring 2023 • Question 2

Question

Math 6040 • Probability Spring 2023 • Question 2

Let \(\{A_{i}\}_{i=1}^{\infty }\) be a sequence of independent events such that \(\sum _{i=1}^{\infty } P(A_{i}) = \infty \). Prove that the \(A_{n}\) occur infinitely often with probability 1 (the converse of the Borel-Cantelli Lemma), i.e. that \(\sum _{i=1}^{\infty }\mathbf {1}_{A_{i}} = \infty \) almost surely.

Raw LaTeX

Let \(\{A_{i}\}_{i=1}^{\infty}\) be a sequence of independent events such that \(\sum_{i=1}^{\infty} P(A_{i}) = \infty\). Prove
that the \(A_{n}\) occur infinitely often with probability 1 (the converse of the Borel-Cantelli
Lemma), i.e. that \(\sum_{i=1}^{\infty}\mathbf{1}_{A_{i}} = \infty\) almost surely.

Topic Tags

Borel-Cantelli Lemmas Classical Probability and Inclusion-Exclusion Convergence in Probability and Almost Sure Convergence Uniform Integrability and Truncation Methods