Math 6010 • Linear Models

Let \((X_n)\) be a sequence of real-valued random variables and define
the tail \(\sigma\)-algebra
\(\mathcal{T}=\cap_{n\geq1}\sigma(X_n,X_{n+1},\ldots)\). Let
\(S_n=X_1+\cdots+X_n\). Prove or disprove the following statements.
\begin{enumerate}
\item[(a)] \(\{\lim_{n\to\infty}S_n\text{ exists}\}\in\mathcal{T}\).
\item[(b)] \(\{\overline{\lim}_{n\to\infty}S_n>0\}\in\mathcal{T}\).
\item[(c)] \(\{\overline{\lim}_{n\to\infty}S_n/c_n>1\}\in\mathcal{T}\) if
\(c_n\to\infty\) as \(n\to\infty\).
\end{enumerate}