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

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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