Math 6010 • Linear Models

Let \((X_n)\) be i.i.d. Cauchy distributed random variables. That is,
they have probability density function \(\frac{1}{\pi(1+x^2)}\). Abbreviate
\(S_n=X_1+\cdots+X_n\).
\begin{enumerate}
\item[(a)] Prove that
\[
  \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}|X_i|=\infty
  \qquad\text{almost surely.}
\]
\item[(b)] Using the fact that the characteristic function of a Cauchy
distributed random variable is given by \(e^{-|t|}\), prove that \(S_n/n\) is again
a Cauchy distributed random variable.
\item[(c)] Deduce that
\[
  \overline{\lim}_{n\to\infty} S_n/n=\infty
  \qquad\text{and}\qquad
  \underline{\lim}_{n\to\infty} S_n/n=-\infty,
  \qquad\text{almost surely.}
\]
\end{enumerate}