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Question

Math 6010 • Linear Models

Spring 2018 • Question 4

Question

Math 6010 • Linear Models Spring 2018 • Question 4

  1. Prove the Borel-Cantelli lemma. That is, prove that if \(\{A_n\}_{n\geq 1}\) is any sequence of events satisfying \(\sum _{n\geq 1}\mathbb {P}(A_n)<\infty \), then \(\mathbb {P}(A_n\text { occurs for infinitely many }n)=0\).
  2. Prove that if \((X_n)\) is any sequence of real-valued random variables, then there are constants \(c_n\) so that \(X_n/c_n\to 0\) almost surely as \(n\to \infty \).

Raw LaTeX

\begin{enumerate}
\item[(a)] Prove the Borel-Cantelli lemma. That is, prove that if
\(\{A_n\}_{n\geq1}\) is any sequence of events satisfying
\(\sum_{n\geq1}\Prob(A_n)<\infty\), then
\(\Prob(A_n\text{ occurs for infinitely many }n)=0\).
\item[(b)] Prove that if \((X_n)\) is any sequence of real-valued random
variables, then there are constants \(c_n\) so that \(X_n/c_n\to0\) almost surely
as \(n\to\infty\).
\end{enumerate}

Topic Tags

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