Alternatives to a one-dimensional Poisson process Say I have "arrival" times in what may or may not be a Poisson process.  I can think of at least three ways in which it can deviate from a Poisson process:


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*Clumping. One arrival is likely to be near other arrivals.  This raises a question of how to estimate where one clump ends and the next begins.  And sizes and shapes of clumps.  It's easy to see one way to reject the hypothesis that it's a Poisson process in favor of this alternative: Kolmogorov–Smirnov, assuming a uniform distribution in time.  How to proceed after rejection is not as clear.

*Repulsion between arrivals close together in time.  It is said that crossovers between chromosomes behave this way.  Maybe times between arrivals could have a gamma distribution with shape parameter $>1$?  Or maybe something else?

*Repulsion between adjacent arrivals regardless of the distance between them.  I have the impression that the distribution of eigenvalues of random Hermitian matrices might be distributed like this, but I don't know what the probability distribution of the gap sizes looks like, let alone how to reject a Poisson process in favor of this alternative.


Right now I'm suspecting I might mainly need to be concerned with the first possibility above.  So what is done, and one is there to read about what is done?
 A: I still not entirely sure what you want, but I’ll give this a try:
If you suspect a process to be Poissonian or be close to it and still describable in terms of arrival probabilities, looking at the distribution of times between arrivals (inter-arrival distribution) is a good approach:


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*For a pure Poissonian process, the inter-arrival distribution is perfectly exponential. In this case, the distribution of waiting times for the next arrival (when starting to wait at random time points) is also exponential (this one may be easier to obtain). Processes that are close to being Poissonian can now be classified by how their inter-arrival distribution deviates from the exponential distribution for the average event-rate of the process (blue in all of the following).


*If there is clumping, arrivals beget arrivals, i.e., if the arrival probability is higher after an arrival, you would expect the inter-arrival distribution (green) to be higher than the exponential distribution (blue) for short inter-arrival times:


*If arrivals repulse each other, the arrival probability is reduced after an arrival and thus the inter-arrival distribution (green) should be lower than the exponential distribution (blue) for short inter-arrival times:


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Repulsion between adjacent arrivals regardless of the distance between them.

Unless you specify how your repulsion works, I would not see a way to distinguish this from a Poisson process with a lower event rate:


*Finally, if your process is nowhere near a Poissonian process, you would expect your inter-arrival distribution to look completely different from an exponential one. For example, if your arrival probability only assumes considerable values during short, regularly occuring time windows, you would get something like this:

Note that if your process is somwhere near a Poissionian one, it may help to think about it in terms of arrival probabilities while thinking of it in terms of arrivals attracting or repelling each other may be confusing. The latter is problematic as it implies that the whole series of arrivals is predetermined and ignores that arrivals are – as a first approximation – independent random events.
