A note on the path to extinction of critical Markov branching processes

For critical Markov branching processes {Zt} with finite variance and with time to extinction T, we obtain the asymptotic behavior of the two processes {ZuT/T,0<=u<1} and {ZT-u,u>0} as Z0=x-->[infinity]. These processes were called by Jagers and co-workers the "path to extinction" and the "verge of extinction" respectively. We show that as Z0=x-->[infinity], (1) for fixed 0<=u<1, ZuT/T converges in distribution to a mixture of two exponential random variables; (2) for fixed u>0, ZT-u converges in distribution, and after normalizing by u-1, the limit further converges to a gamma distribution as u-->[infinity]. Simulation is used to illustrate the results.