  Consider the continuous-time Markov Branching Process. In critical case we
consider a situation when the generating function of intensity of
transformation of particles has the infinite second moment, but its tail
regularly varies in sense of Karamata. First we discuss limit properties of
transition functions of the process. We prove local limit theorems and
investigate ergodic properties of the process. Further we investigate limiting
probability function conditioned to be never extinct. Hereupon we obtain a new
stochastic population process as a continuous-time Markov chain called the
Markov Q-Process. We study main properties of Markov Q-Process.
