Asymptotics of probabilities of large deviations for critical Markov branching processes (Q789089)

Let \(\mu\) (t), \(t=0,1,2,...\) be a Galton-Watson process, \(\mu(0)=1\). Denote \(F(s)=\sum^\infty_{k=0}P[\mu(1)=k]s^ k\) and assume that F(s) is an analytic function in the disc \(| s|<1+\epsilon\) for some \(\epsilon>0\). Furthermore, let \(F'(1)=1\), \(b=F''(1)>0\), \(c=F'(1)\), \(\theta =2c/(3b^ 2)-1\), \(\ln_ 1 t=\ln t\) and \(\ln_{j+1} t=\ln \ln_ j t\) for \(j=1,2,...\). If \(0<x=o(t/\ln_ N t)\) as \(t\to \infty\) for some \(N>2\), then \[ \exp [x(1-2\theta \ln t/t)]P\{\mu(t)>xE[\mu(t)| \mu(t)>0]| \mu(t)>0\}\to 1. \] Moreover, if the maximal jump of the process \(d=1\) and \(n=o(t^ 2/\ln_ N t)\) as \(t\to \infty\), then \[ P[\mu(t)=n]\sim(2/bt)^ 2\exp \{-(1- \theta \ln t/t)2n/bt\}. \] The work is a continuation of the author's investigations in Teor. Veroyatn. Primen. 25, 490-501 (1980; Zbl 0436.60058).