Let $N(t)$ be a renewal process, with a sequence of IID inter-arrival times $X_{1}, X_{2}, \dots$ having finite second moment: $EX_{i}^{2} < \infty$.
Then we know that $$\mathrm{Var}N(t)= 2 \int^{t}_{0} m(t-s) \cdot m'(s)ds + m(t) - m(t)^{2}$$
where $m(t) = E[N(t)]$ and can be written $m(t) = F(t) + \int_{0}^{t} m(t-x)f(x)dx$ where $f$ is the density of the inter-arrival times and $F$ is the CDF.
How can I compute limit $$ \lim_{t\to\infty}\frac{\mathrm{Var}N(t)}{t} $$ I guess, this could be done with Laplace transform, but there is this $\frac{1}{t}$ I can't get rid of.