Math 6010 • Linear Models

Let \(Z_1,Z_2,\ldots\) be iid with
\(\Prob(Z=1)=\Prob(Z=-1)=1/2\), and let \(c_1,c_2,\ldots\) be given constants.
\begin{enumerate}
\item[(a)] Express the characteristic function of \(\sum_{k=1}^{n}c_kZ_k\) in
terms of standard elementary functions.
\item[(b)] Show that \(\sum_{k\geq1}c_kZ_k\) converges almost surely
\(\Longleftrightarrow\) \(\sum_{k=1}^{\infty}c_k^2<\infty\). For this you can use
results we proved in class, or other problems on this exam.
\end{enumerate}