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It's not hard to show that the exponential distribution is the only inter-arrival distribution with a constant event rate. And if you consider the distribution of the number of events falling into an interval, you get a Poisson distribution, with the overall process being a Poisson process. So, it followed that the Poisson point process should be the only point process with a constant event rate. But then I thought about a simple compound Poisson point process. To keep things simple, assume that every arrival of the Poisson process, five events happen instead of just one event. Let's call this a deterministically compounded Poisson process. It seems that this process should also have a constant event rate. The expected number of events in any interval should still be independent of any other interval. And if this deterministically compounded point process has a constant event rate, then other kinds of compounding, where the number of events happening at every arrival is some random variable independent of the underlying Poisson process should also have a constant event rate. Can we say then that a Poisson process along with any compound Poisson process and nothing else should have a constant event rate?

Rohit Pandey
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    "Memoryless constant event rate" – user619894 Sep 29 '21 at 06:01
  • At least there is this subtlety of definition: If a point process creates a random set of points, you cannot have more than one point at the same time. – Jukka Kohonen Sep 29 '21 at 18:01
  • @JukkaKohonen - that would resolve this apparent paradox. What does a "set of points" mean? Any references for this? – Rohit Pandey Sep 29 '21 at 19:32
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    Well, just what a set usually means (as opposed to, say, multiset). I'm sure that if you look at various references about point processes, they will contain different language; some may say "set", some "collection of points" etc. For just one example, this MIT course https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/course-notes/MIT6_262S11_chap02.pdf explicitly says the arrival times of an "arrival process" are strictly increasing. – Jukka Kohonen Sep 29 '21 at 19:58

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