On a Markov analogue of continuous-time \(Q\)-processes (Q2849245)

Let \((Z_t,\operatorname{P})\) denote a continuous-time, one-type Markov branching process. For any real \(\tau\geq 0\), define the transition function NEWLINE\[NEWLINEQ_{ij}(t):= \lim_{r\to\infty}\operatorname{P}(Z_{t+\tau}= j\mid Z_t= i,\, Z_{t+\tau+r}> 0).NEWLINE\]NEWLINE By studying the asymptotic behavior of the generating function NEWLINE\[NEWLINEG_i(t,x):=\sum_j Q_{ij}(t)x^j,NEWLINE\]NEWLINE as \(t\to\infty\), the author obtains asymptotic expressions for \(Q_{ij}(t)\), \(t\to\infty\), in cases of subcritical, critical, as well as supercritical \((Z_t,\operatorname{P})\).