Show that a simple point process N on R+ is a Poisson process with rate λ if and only if N has independent increments, E[N(1)] = λ, and, for any t and n, the conditional joint density of T1,...,Tn...

Show that a simple point process N on R+ is a Poisson process with rate λ if and only if N has independent increments, E[N(1)] = λ, and, for any t and n, the conditional joint density of T1,...,Tn given N(t) = n is the same as the density of the order statistics of n independent uniformly distributed random variables on [0, t].