Math 6010 • Linear Models

Let \(X_1,X_2,\ldots\) be iid random variables with common pdf
\[
  f(x)=
  \begin{cases}
    0, & |x|\leq2,\\
    \dfrac{c}{x^2\log|x|}, & |x|>2.
  \end{cases}
\]
\begin{enumerate}
\item[(a)] Prove that there is a finite constant \(c\) which makes \(f\) a pdf.
\item[(b)] Let \(S_n=X_1+X_2+\cdots+X_n\). Prove that \(S_n/n\to0\) in probability
as \(n\to\infty\). \textbf{Hint:} Prove that \(S_n/n\) converges to 0 in
distribution, then explain why you can upgrade to convergence in probability
for free.
\item[(c)] Does \(S_n/n\to0\) almost surely as \(n\to\infty\)?
\end{enumerate}