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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 2

Question

Math 6010 • Linear Models Spring 2020 • Question 2

Let \(A_1,A_2,\ldots \), be a sequence of events (not necessarily independent) such that \[ \sum _{n=1}^{\infty }\mathbb {P}(A_n)=\infty \qquad \text {and}\qquad \mathbb {P}(A_n\cap A_m)\leq \mathbb {P}(A_n)\mathbb {P}(A_m) \quad \text {for } m\ne n. \] Prove that \(\mathbb {P}(A_n\text { i.o.})=1\) or provide a counterexample. Hint: Consider the mean and variance of \[ \sum _{i=1}^{n}\mathbf {1}_{A_i}. \]

Raw LaTeX

Let \(A_1,A_2,\ldots\), be a sequence of events (not necessarily
independent) such that
\[
  \sum_{n=1}^{\infty}\Prob(A_n)=\infty
  \qquad\text{and}\qquad
  \Prob(A_n\cap A_m)\leq\Prob(A_n)\Prob(A_m)
  \quad\text{for } m\ne n.
\]
Prove that \(\Prob(A_n\text{ i.o.})=1\) or provide a counterexample.
\textbf{Hint:} Consider the mean and variance of
\[
  \sum_{i=1}^{n}\mathbf{1}_{A_i}.
\]

Topic Tags

Analysis of Variance Borel-Cantelli Lemmas Covariance and Independence Uniform Integrability and Truncation Methods