Математика институти

The scientific novelty of the research is as follows:

asymptotic representations for generating functions and their differential of
noncritical branching processes with infinite moment of order 1 + ε , ε > 0 , with
discrete and continuous time are obtained; explicit forms of generating functions of invariant measures of the processes are obtained;
an asymptotic representation for the transition probabilities of the critical Galton-Watson processes with immigration, forming an ergodic Markov chain is obtained, in which the immigration stream law has an infinite first moment and the per capita offspring law has an infinite second moment;
the Basic lemma and its differential analogue of the theory of critical Galton- Watson Branching Process and continuous time Markov Branching Processes with infinite second moment are proved; the speed rates of convergence to invariant

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measures in Galton-Watson Branching Process and in continuous time Markov Branching Processes with immigration are established; explicit forms of the generating functions of invariant measures of these processes are found;

the Laplace transform for the limit of the joint distribution law of population size and the total progeny in Q-process is found; the Law of Large Numbers and the Central Limit Theorem for the total progeny in Q-process are proved; components of the q-matrix of the Markov Q-processes are found, structural and asymptotic states of Markov Q-processes are classified.