Let {N(t): t≥0} be a nonhomoegneous Poisson process with continuous rate function λ(t)>0. Let T1<T2<... denote the times of the points. Show that
E( |Tn-1| |N(1)=n)->1 as n->∞
For nonhomoegneous Poisson process, I know that we have independent increments and that,
Number of points in (t,t+s]
=N(t,t+s] is Poisson distributed with mean
t+s
∫ λ(u)du
t
But do I have to use this? I have no idea where to start and I would appreicate if someone can help.
Thank you!
OK, but how should I begin? Do I need to find the conditional probability density function of |Tn-1| |N(1)=n?
( and ) iff ( and )
I can see why this is true, but with this we still cannot use the independent increments property becuase the intervals are overlapping, right?
Thank you!