This is probably well known in Renewal theory. But I didn't find any proof.
Let $N(t)$ be the renewal process associated with $X_0=1$ and $X_1, X_2, \ldots$ iid $F$ with mean $\mu$ and variance $\sigma^2$. Then $$\frac{\operatorname{Var}(N(t))}{t} \to \frac{\sigma^2}{\mu^3}$$
Let $E(N(t))=U(t)=1+\sum\limits_{n=1}^\infty F^{*(n)}(t)$. After playing with $E(N(t))^2$ I got that
$$E(N(t))^2=2U*U(t)-U(t)$$
I am stuck here. The problem is that known limit theorems do not help unless we have direct Riemann integrability and some other stuff.