I understand that one of the basic assumptions of a Poisson process is that in a small enough interval, the probability of more than one arrival is negligible.
In my case, I have data that shows the number of arrivals per day. Most days contain no arrivals, some have 1, few have 2, but there is a single occasion where there are 10 arrivals in one day. Since I don't have the precise times at which these 10 arrivals occurred, this doesn't necessarily violate one of the Poisson assumptions that I stated above, does it?
How should I treat this situation? If I fit a Poisson distribution to the count data, then that anomaly day will cause $\lambda$ to be larger than it would otherwise. On the other hand, that 10-day is rare, but also somewhat explainable given where the data comes from. I.e. I think the arrivals follow a Poisson process, but occasionally they will come in a "cluster".
Can you please give me some advice on how these situations are usually dealt with? I feel there are really two questions lurking underneath here: one having to do with discretizing data, and the other to do with compound Poisson processes. But these two concepts are blurred together for me.