I am looking for a reference on Poisson processes that study the probability distribution of the random times of jumps of Poisson processes. I am particularly interested by the following result, that I believe to be true.
Proposition
Let $(X_t)_{t \geq 0}$ be a Poisson process with $0 \leq t_1 \leq \cdots \leq t_n \leq \cdots$ its random times of jumps and $N$ its random number of jumps on $[0,1]$.
Then, conditionally to $N=n \geq 1$, the random vector $(t_1, \ldots , t_n)$ has the same law that the ordered statistics of a vector $(U_1,\ldots, U_n)$ of i.i.d. uniform random variables on $[0,1]$.
Any help would be appreciated.