Assume $X_1,X_2,\ldots$ are positive inter-arrival times of a renewal process with pdf $p(x)$, i.e., the $k^\text{th}$ arrivals occurs at $\sum_{i=1}^k X_k$. What is the pdf of inter arrival times given the $N(t)=n$, i.e., the $n^\text{th}$ arrival occured at time $t$? Show that this PDF converges to the $p(x)$.
For Poisson, I know the conditional arrival times are uniformly distributed in $[0,t]$, but I am looking for conditional inter-arrival times. And, the same thing for a renewal process.
This should have been well studied in the literature. Can you point me to the right reference, or show me how to prove this? Giving the answer for Poisson point process would be helpful too.