Math 6010 • Linear Models

Suppose \(X_1,X_2,\ldots\) are independent mean-zero random variables
that satisfy the condition
\[
  \sum_{i=1}^{\infty}\Var(X_i)<\infty.
\]
\begin{enumerate}
\item[(a)] Prove that \(\sum_{n=1}^{\infty}X_n\) exists a.s. and is finite a.s.
\item[(b)] Use part (a) to prove that if \(\varepsilon_1,\varepsilon_2,\ldots\)
are i.i.d. with \(\Prob\{\varepsilon_1=\pm1\}=1/2\), then the random harmonic
series \(\sum_{i=1}^{\infty}(\varepsilon_i/i)\) converges a.s. and is finite a.s.
\item[(c)] Is \(\sum_{i=1}^{\infty}(\varepsilon_i/i)\) in \(L^1(\Prob)\)?
\end{enumerate}