Math 6010 • Linear Models

Let \(X_1,X_2,\ldots\) be independent random variables with distribution functions \(F_1,F_2,F_3,\ldots\). Let \(Y=\sup_{1\leq i<\infty}X_i\).
\begin{parts}
\item Show that \(P\{Y<\infty\}\) is 0 or 1 depending on whether \(\sum_{1\leq i<\infty}(1-F_i(x))\) converges for some \(x\). (5 points)
\item Show that if \(P\{Y<\infty\}=1\), then \(\prod_{1\leq i<\infty}F_i(x)\) converges for all \(x\) and it is the distribution function of \(Y\).
\end{parts}