My book says that if we know that if we are viewing a poisson process with length $t$, and know that $n$ events happened in that interval, than the time that any of those events happened is uniformly distributed with probability $1/t$, I do not doubt that this is true, but I am wondering if the logic my book uses is correct.
They say that what I wrote above must hold because of this theorem:
Now, I am wondering why the logic holds. What I mean is:
Lets say we know that all the events are uniformly distributed, then we know that the $S$s must have the distribution of the order statistics. And hence we have the result above, so the result theorem above become a necessary condition.
But conversely, let's say we know that the $S$s have the distribution of the order statistics of $n$ independent random variables uniformly distributed on $[0,t]$. Can we then say that the random variables the order statistics represent is uniform on $[0,t]$, and hence the above theorem is a sufficient condition for what I wrote first?
Maybe(?) another way of asking the question is: I guess it is pretty clear that the order statistics are described clearly by the random variables?, but conversely if we have the order statistics, is it clear what kind of random variables we are dealing with?