Renewed Limit Theorems for the discrete-time Branching Process and its Conditioned Limiting Law interpretation

TL;DR

This paper advances the understanding of discrete-time Galton-Watson branching processes by establishing new limit theorems, analyzing the Q-process, and providing joint and asymptotic laws for population trajectories.

Contribution

It introduces a differential analogue of the Basic Lemma and improves classical results, offering new insights into the asymptotic behavior of the Q-process and its total state.

Findings

Derived a joint limit law for Q-process and total state

Proved analogue of Law of Large Numbers for total state

Established Central Limit Theorem for total state

Abstract

Our principal aim is to observe the Markov discrete-time process of population growth with long-living trajectory. First we study asymptotical decay of generating function of Galton-Watson process for all cases as the Basic Lemma. Afterwards we get a Differential analogue of the Basic Lemma. This Lemma plays main role in our discussions throughout the paper. Hereupon we improve and supplement classical results concerning Galton-Watson process. Further we investigate properties of the population process so called Q-process. In particular we obtain a joint limit law of Q-process and its total state. And also we prove the analogue of Law of large numbers and the Central limit theorem for total state of Q-process. Keywords: Branching process; transition function; Q-process; invariant measures; ergodic chain; total states; joint distribution; limit theorem.