Assume $\{X_i:i=1,2,\ldots\}$ are independent random variables with probability distribution $f(x)$, associated with a renewal process. If we know the number of renewals at time $t$ is $n$, find the join probability distribution of $X_i$s when $t\to \infty$.
Solution: First, we should find $$\mathbb{P}\left(X_1\leq x_1, \ldots, X_n\leq x_n \Bigg| N(t)=n\right) =\mathbb{P}\left(X_1\leq x_1, \ldots, X_n\leq x_n \Bigg| \sum_{k=1}^{n}X_k \leq t < \sum_{k=1}^{n+1}X_k\right) = \frac{\mathbb{P}\left(X_1\leq x_1, \ldots, X_n\leq x_n \bigcap \sum_{k=1}^{n}X_k \leq t < \sum_{k=1}^{n+1}X_k\right)}{\mathbb{P}\left( \sum_{k=1}^{n}X_k \leq t < \sum_{k=1}^{n+1}X_k\right)}$$
I have no idea how to continue from here. I think this should have been addressed in the literature. Please let me know your thoughts.