1. ## nonhomoegneous Poisson process

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!

2. Originally Posted by kingwinner
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 equal to
t+s
∫ λ(u)du
t
No, the number of points in (t,t+s] is a Poisson random variable whose parameter is the integral you wrote.

About your question. You should try to use these facts (and similar ones):
- $\displaystyle T_n<1$ when $\displaystyle N(1)=n$
- ($\displaystyle 1-\varepsilon<T_n$ and $\displaystyle N(1)=n$) iff ($\displaystyle N(1-\varepsilon,1)\geq 1$ and $\displaystyle N(0,1)=n$).

3. Originally Posted by Laurent
About your question. You should try to use these facts (and similar ones):
- $\displaystyle T_n<1$ when $\displaystyle N(1)=n$
- ($\displaystyle 1-\varepsilon<T_n$ and $\displaystyle N(1)=n$) iff ($\displaystyle N(1-\varepsilon,1)\geq 1$ and $\displaystyle N(0,1)=n$).
OK, but how should I begin? Do I need to find the conditional probability density function of |Tn-1| |N(1)=n?

($\displaystyle 1-\varepsilon<T_n$ and $\displaystyle N(1)=n$) iff ($\displaystyle N(1-\varepsilon,1)\geq 1$ and $\displaystyle N(0,1)=n$)
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!