  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.
