Let $(N_t)$ be a standard Poisson process. By construction we know that if $(T_i)$ are the jump times then $T_{i+1}-T_i$ are independent and identically distributed exponential random variables. It is also well know that conditionally on the set $\{N_1 = n\}$ (after a unit of time the process had $n$ jumps) then the jump times $(T_i)$ are distributed as the order statistic $(U_{(i)})$ of $n$ independent and uniform random variables.
My question: Is it true that the differences $U_{(i)}-U_{(i-1)}$ are also i.i.d.?
I think is not to hard to probe that they have the same distribution but i don't know if is it true that they are independent.
Any help will be appreciated
(Corrected to include the last difference.) The differences are exchangeable: Define $V_i:=U_{(i)}-U_{(i-1)}$ for $i=1,2,\ldots,n$ ($U_{(0)}:=0$) and $V_{n+1}:=1-U_{(n)}$, and let $\sigma$ be any permutation of $\{1,2,\ldots,n+1\}$. Then $(V_{\sigma(1)},V_{\sigma(2)},\ldots,V_{\sigma(n+1)})$ has the same distribution as $(V_1,V_2,\ldots,V_{n+1})$.