The renewed limit theorems for the discrete-time branching process and its conditioned limiting law interpretation

Year 2016, Volume: 4 Issue: 4, 213 - 238, 31.12.2016

Azam Abdurakhimovich Imomov

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

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