Coordinate-wise Poisson Point Process

Question

Let $(X_i)_{i=1}^\infty$ and $(Y_i)_{i=1}^\infty$ be independent Poisson point processes of intensity $\lambda$ and $\delta$ on $[0,\infty)$ respectively, ordered in increasing order (hence, they are the ordered jump times of independent Poisson processes of the corresponding intensities). Define a new point process $(Z_i)_{i=1}^\infty$ on $[0,\infty)^2$ via $Z_i = (X_i,Y_i)$.

Is $(Z_i)$ a(n inhomogeneous) Poisson process on $[0,\infty)^2$? If so, what is the corresponding intensity function?

Answer 1

Your intuition about order is correct, and you do not get independence of distinct areas.

For example, if you have any points in the rectangle $(a,a+\lambda]\times(b-\delta,b]$ then the conditional probability of having any points in $(a-\lambda,a]\times(b,b+\delta]$ is zero, since it would require some $X_{i+1}-X_{i}$ or some $Y_{i+1}-Y_{i}$ to be negative, contrary to the construction that both sequences are ordered.

Here is a simulation of $100$ cases in R, one case illustrated with a line which clearly takes increasing steps (you never go "right and down" or "left and up") and the other $99$ with points to suggest density (highest near $0$ and higher near the diagonal than off the diagonal):

set.seed(2023)
plot(cumsum(rexp(1000,1)), cumsum(rexp(1000,1)), xlab="", ylab="", 
     xlim=c(0, 100), ylim=c(0, 100), type="l")
for (i in 2:100){
  points(cumsum(rexp(1000,1)), cumsum(rexp(1000,1)), pch=".", col=i)
  }