Learn English – Difference between “numbers analysis” and “number analysis”

In terms of the verb agreeing with the number, you need to be clear about what is doing the verb. For example, "A number of people are waiting for the bus", it is the people who are waiting, and there is more than one person, so use are. Here using "a number" could be replaced by "some". However, when it is the number that is doing the verb, eg, increasing, then it should be singular. Eg. The number of users has increased. But what about "the numbers of users...". Here you would use are, but why would numbers be plural? Are we talking about more than one groups of users?

Since you refer to each (singular) sub-interval, number should also be singular. Assuming that an independent number makes sense in your field of study, you could say:

The number of events occurring in each sub-interval is independent.

If the word independent requires two arguments (X is independent of Y), you could say:

In each sub-interval, the number of events occurring is independent of that of any other sub-interval.

It may be better to use multiple but simpler sentences to express your final sample sentence:

Let X_i be the number of events occuring in sub-interval i. Then the X_i are stochastically independent.