Asymptotic behavior for supercritical branching processes.

Let { Z (t) ; t ⩾ 0 } be a Markov branching process (MBP). There exists a well-known sequence { C (t) ; t ⩾ 0 } such that W (t) ≔ Z (t) / C (t) a.s. converges to a non-degenerate random variable W as t → ∞. This paper attempts to study the asymptotic behavior of P (Z (t) = k t) and P (0 ⩽ Z (t) ⩽ k t) with k t e − λ t → 0 as t → ∞ for MBPs, which helps to study large deviations of Z (t + s) / Z (t). Moreover, we obtain the local limit theorem of this process as an additional finding. During the argumentation, the Cramér method is applied to analyze the large deviation of the sum of random variables. [ABSTRACT FROM AUTHOR]