Math 6010 • Linear Models

Suppose \(X_1,X_2,\ldots\) are iid with \(\E|X_i|=\infty\). Let
\(S_n=X_1+\cdots+X_n\), and let \(A\) be the event that \(M_n=S_n/n\) converges to a
finite limit. Let \(B\) be the event that \(|X_n|\geq n\) infinitely often.
\begin{enumerate}
\item[(a)] State the definition of \(B\) in terms of unions and intersections.
\item[(b)] Show that \(\Prob(B)=1\).
\item[(c)] Verify the identity
\[
  M_n-M_{n+1}=\frac{M_n}{n+1}-\frac{X_{n+1}}{n+1}
\]
and use it to show that \(A\cap B=\emptyset\).
\item[(d)] Use the above to prove that \(\Prob(A)=0\).
\end{enumerate}