@article{article_526521, title={The renewed limit theorems for the discrete-time branching process and its conditioned limiting law interpretation}, journal={New Trends in Mathematical Sciences}, volume={4}, pages={213–238}, year={2016}, url={https://izlik.org/JA65ZN37PX}, author={Imomov, Azam Abdurakhimovich}, keywords={Branching process,transition function,Q-process,invariant measures,ergodic chain,total states,joint distribution}, abstract={<p class="MsoNormal" style="text-align:justify;text-justify:inter-ideograph; text-indent:36.0pt;tab-stops:center 240.0pt right 475.0pt">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. <span style="font-family:"Times New Roman","serif";mso-no-proof:yes"> <o:p> </o:p> </span> </p>}, number={4}