Refinement of the Main Lemmas of the Theory of Critical Branching Processes

In the paper, we consider critical Markov branching random processes of continuous time and branching random processes of discrete time (critical Galton–Watson processes) defined respectively by the generating functions $$F(t,x)$$ and $$F_{n}(x)$$ $$(n=0,1,\cdots,\;|x|\leq 1).$$ In this case, the generating function $$F(t,x)$$ will be a solution to an ordinary differential equation, the right side of which is a nonlinear function of $$F(t,x),$$ and the function $$F_{n}(x)$$ is equal to the number of descendants of one particle at the $$n$$ -th iteration of the generating function. Asymptotic analysis of generating functions $$F(t,x)$$ and $$G_{n}(x)$$ for $$t\to\infty,$$ $$n\to\infty,$$ respectively, plays a major role in solving the main problems of the theory of branching random processes. Statements related to the asymptotic analysis of generating functions $$F(t,x)$$ and $$F_{n}(x)$$ for $$t\to\infty,$$ $$n\to\infty,$$ respectively, came to be called the main lemmas.

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