# limiting variance in Renewal Theory

Let $\{X_n\}_{n\ge 0}$ be a renewal process and define $S_n=\sum_{i=0}^{n}X_i$ and $N(t)=\sum_{n=1}^\infty\mathbb{1}[S_n\le t]$. Let $E(X_i)=\mu$ and $\text{Var}(X_i)=\sigma^2$. Let $V(t)=\text{Var}(N(t))$. Show that $\lim_{t\to\infty}\frac{V(t)}{t}=\frac{\sigma^2}{\mu^3}$.