Math 6010 • Linear Models

Let \((X_n)\) be i.i.d. random variables with mean \(\mu\).
\begin{enumerate}
\item[(a)] Prove that
\[
  \lim_{n\to\infty}
  \frac{1}{\log n}\sum_{i=1}^{n}\frac{X_i}{i}
  =
  \mu
  \qquad\text{almost surely.}
\]
\item[(b)] If the random variables also have a finite variance
\(\sigma^2<\infty\), then find \(a_n\) such that
\[
  a_n\left(\sum_{i=1}^{n}\frac{X_i}{i}-\mu\log n\right)
\]
has a weak limit. Identify this weak limit. (You need to prove the weak
convergence, once you have identified \(a_n\) and the limiting distribution.)
\end{enumerate}

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