Central Limit Theorem and Law of Large Numbers Analogues for the Total Progeny in the Q-Processes

Abstract

In this paper we are dealing with population size growth systems called Q-processes. The class of trajectories of these systems is a subset of the family of all possible trajectories of the ordinary Galton-Watson branching system, provided that they do not decay in the remote future. We observe the total progeny of a single founder-individual, generated by the reproduction law of the Q-process up to time n. By analogy with branching systems models, this variable is of great interest in studying the deep properties of the Q-process. Our main results are analogues of Central Limit Theorem and Law of Large Numbers for Sn, denoting the total progeny of a single founder-individual, generated by the reproduction law of the Q-process up to time n. We find that the total progeny as a random variable approximates the standard normal distribution function under a second moment assumption for the initial Galton-Watson system offspring law. We estimate the speed rate of this approximation. We also prove an analogue of the law of large numbers with an estimate of the approximation rate to the degenerate distribution.