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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 $X_t$?

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And how to show the limits exist? – epsilon Nov 13 '11 at 23:35
The KTF question: What do you know? What have you tried? Where did you fail? – Did Nov 16 '11 at 15:23

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.

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The limit of the variance of $N_t/t$ when $t\to\infty$ is zero, not $\sigma^2/\mu^3$. – Did Jul 26 '12 at 14:24

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