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.