I am looking for a book/paper that studies questions as the following for branching processes:
The probability that there are at least X individuals in the branching process after t steps is at least y.
I usually only find theorems stating asymptotic behaviour for t approaching infinity. I want to ask the question for a given fixed t (and fixed X).
I even have the setting that there is a finite set of nonzero branching probabilities and I am not sure whether this helps.
Someone must have studied a similar question.