Math 6010 • Linear Models

Let \(X_1,X_2,\ldots\) be independent identically distributed positive random variables with \(EX_1=\mu\) and \(\operatorname{var}(X_1)=\sigma^2\). Let
\[
N(t)=\min\left\{k:\sum_{1\leq i\leq k}X_i>t\right\},\qquad 0<t<\infty.
\]
\begin{parts}
\item Show that
\[
\lim_{t\to\infty}\frac{1}{t}N(t)=\frac{1}{\mu}
\]
almost surely. (5 points)
\item Show that \((N(t)-t/\mu)/(t\sigma^2/\mu^3)^{1/2}\) goes in distribution to a standard normal random variable. (5 points)
\end{parts}