A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications

Abstract

We obtain a differential analog of the main lemma in the theory of Markov branding processes $\mu(t),\quad t \geq 0$, of continuous time. We show that the results obtained can be applied in the proofs of limit theorems in the theory of branching processes by the well-known Stein - Tikhomirov method. In contrast to the classical condition of nondegeneracy of the branching process $\{\mu(t) > 0\}$, we consider the condition of nondegeneracy of the process in distant $\{\mu(\infty) > 0\}$ and justify in terms of generating functions. Under this condition, we study the asymptotic behavior of trajectory of the considered process.