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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.

Goulifet
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  • I may miss something, but the $t_i$'s can be greater than 1, $U_i$'s cannot. How can they have the same law? – Jay.H Nov 19 '15 at 13:13
  • My question is conditionally to $N = n$, with $N$ the number of jumps between $0$ and $1$. The event ${N=n}$ is equal to ${t_n \leq 1 < t_{n+1}}$ (I remind that the $t_n$ are ordered). In particular, the $t_i$ are smaller than $1$ for $i \leq n$. – Goulifet Nov 20 '15 at 10:35
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    I guess the following post is relevant http://math.stackexchange.com/q/1529324/289489 – Jay.H Nov 20 '15 at 13:11
  • Definitely relevant. Thanks a lot. – Goulifet Nov 21 '15 at 16:45

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