For an Orderly Poisson Process, events occur at distinct points and not simultaneously. However, the reverse is not necessarily true, i.e, even if the events occur at distinct points, the process may not be orderly.
Can you please explain why this happens? Why distinct occurrences do not guarantee orderliness? It will be really helpful if you can clear this conundrum with an example.
Thank you! :)
Example: An excursion of a Brownian motion $(B_t: t\ge 0)$ is a piece $(B_t: a\le t\le b)$ of its path such that $B_a=B_b=0$ but $B_t\not=0$ for all $t\in(a,b)$. Let us use the left endpoint of the time interval of a Brownian excursion as a label for the excursion. The collection of these left endpoints is a point process on $(0,\infty)$ comprising countably many points. Different excursion have distinct labels. But this point process is not orderly: between any two of these left endpoints there are infinitely many more such left endpoints.