Conditional law of the arrival times of a Poisson process

Question

The following is Exercise 2.1.(i) from the book "Introductory Lectures on Fluctuations of Levy Processes with Applications". I use the book for self study and couldn't prove this exercise. I would be very happy if someone could give me an idea about the first two steps. This is the exercise:

Let $N=\{N(t):t\geq 0\}$ be a Poisson process with parameter $\lambda$ and let $\{T_i:i\geq 0\}$ be the arrival times, i.e. $T_i:=\inf\{t:N(t)=i\}$. Furthermore $\{X_i:i\geq 0\}$ be the interar-rival times, i.e. $X_{i+1}:=T_{i+1}-T_i$. By recalling that the inter-arrival times are independent and exponential distributed, show that for any $A\in\mathcal{B}([0,\infty)^n)$, $$ P((T_1,...,T_n)\in A|N_t=n) =\int_A \frac{n!}{t^n}1_{(0\leq t_1\leq...\leq t_n\leq t)}dt_1\times\cdots dt_n. $$

So what I can see is that should be enough to prove the claim for $A=(-\infty,a_1]\times\cdots\times(-\infty,a_n]$ and to use the independency I guess we should write the left hand side with the inter-arrival times. Could someone suggest how to start?

Answer 1

First, you can use the Jacobian method to find the joint density of $(T_1,\ldots,T_{n+1})$. Obtain \begin{align*} f_{T_1,\ldots,T_{n+1}}(t_1,\ldots,t_{n+1}) &= f_{X_1,\ldots,X_{n+1}}(t_1,t_2-t_1,\ldots,t_{n+1}-t_{n}) \\ &= \lambda\exp(-\lambda t_1)I(t_1 \geq 0)\prod_{i=2}^{n+1}{\lambda\exp(\lambda(t_{i-1}-t_{i}))I(t_i-t_{i-1}\geq 0)} \\ & \lambda^{n+1}\exp(-\lambda t_{n+1})I(0 \leq t_1 \leq t_2 \ldots \leq t_{n+1}) \end{align*} Next, note that \begin{align*} P((T_1,\ldots,T_n) \in A|N_t=n) &= \frac{P((T_1,\ldots,T_n) \in A, N_t=n)}{P(N_t=n)} \\ &= \frac{P((T_1,\ldots,T_n) \in A, T_{n+1} > t)}{P(N_t=n)} \\ &= \frac{\int_{A \times (t,\infty)}{f_{T_1,\ldots,T_{n+1}}(t_1,\ldots,t_{n+1})d\textbf{t}}}{\frac{\exp(-\lambda t)(\lambda t)^n}{n!}} \\ &= \frac{\int_{A \times (t,\infty)}{\lambda^{n+1}\exp(-\lambda t_{n+1})I(0 \leq t_1 \leq t_2 \ldots \leq t_{n+1})d\textbf{t}}}{\frac{\exp(-\lambda t)(\lambda t)^n}{n!}} \\ &= \int_{A}{\frac{n!}{t^n}I(0 \leq t_1 \leq \ldots t_n \leq t)d\textbf{t}} \end{align*}