Why are non-homogenous poisson processes not independent?

Question

Why are non-homogenous poisson processes not independent?

Let $ N (t)$ be a non-homogeneous Poisson process with intensity $\lambda t$, t > 0. Find the joint density of the first two sojourn times, and deduce that they are not in general independent.

Answer 1

Since you haven't shown your own thoughts on this question, I can only give you hints.

Since you are given an intensity function, use it to compute explicitly the distribution of the number of events at time $t$. Then use this information to find the distribution of the first event time, and then the second event time. Then compute the joint density of the first two event times, and show it is not equal to the product of the marginal densities.

Answer 2

$N(t)$, the number of events up to time $t$, is Poisson with parameter $\int_0^t \lambda s\; ds $. The CDF of the first sojourn time $T_1$ is $F_{T_1}(t) = P(N(t) \ge 1) = 1 - P(N(t) = 0) $. The density for the first sojourn time is the derivative of this.

Given the first event occurs at time $t_1$, the number $N(t_1+t_2) - N(t_1)$ of events from time $t_1$ to time $t_1+t_2 $ (where $t_2 > 0$) is Poisson with parameter $\int_{t_1}^{t_1+t_2} \lambda s\; ds$. Thus the conditional CDF of the second sojourn time, given the first was $t_1$, is $F_{T_2|T_1}(t_2|t_1) = 1 - P(N(t_1 + t_2) - N(t_1) = 0)$, and the conditional density is the derivative of that with respect to $t_2$.

From these you can compute the joint density. You'll find it's not a product.

As for the "why": the basic idea is that if the first event occurs at a time when the intensity is higher, the second event is more likely to happen soon after.