Random Sums of Independent Indicators and Generalized Reduced Processes

We consider a random sum of independent and identically distributed Bernoulli random variables. We prove several limit theorems for this sum under some natural assumptions. Using these limit theorems a generalized version of the reduced critical Galton-Watson process will be studied. In particular we find limit distributions for the number of individuals in a given generation the number of whose descendants after some generations exceeds a fixed or increasing level. An application to study of the number of “big” trees in a forest containing a random number of trees will also be discussed.