moments of Renewal function

Question

Given a renewal process ${X_t}$. How to prove that $\lim_{t\rightarrow \infty}{E[\left(N(t)/t\right)^2]}<\infty$? Does one also have $\lim_{t\rightarrow \infty}{E[\left(N(t)/t\right)^4]}<\infty$ or do we need additional assumption on $X_t$?

Answer 1

the variance of N(t) is standard, Ross' probability models alludes to it without actually doing it, but I'm sure it is in Feller v. 2 ( it is Var$(\frac {N(t)} t )\rightarrow \frac {\sigma^2}{\mu^3}$), and is done using more renewal theory. $\frac {N(t)}t$ will certainly have moments of all orders, which you can see by just comparing it to what a bernouilli does.