Critical Markov Branching-Immigration Systems: Complete Classification of the State Space and Exact Rates of Convergence

Source abstract

We study a critical homogeneous continuous-time Markov branching-immigration system whose offspring and immigration generating functions are regularly varying at the point~1 of orders $1+\nu$ and $\delta$ with $\nu,\delta\in(0,1]$, the slowly varying factors being controlled through their index functions. Two numbers govern the classification: the difference $\gamma=\delta-\nu$ of the two orders and the limit $\Lambda_{\infty}$ of the ratio of the immigration factor to the offspring one. We classify the state space completely: positive recurrent for $\gamma>0$, transient for $\gamma< 0$, and on the boundary $\gamma=0$ null recurrent or transient according as $\Lambda_{\infty}\le{\nu}$ or $\Lambda_{\infty}>\nu$, which at $\nu=\delta=1$ reduces to the classical threshold. In each regime we identify the limiting generating function of the normalised transition functions together with the exact rate of convergence to it, uniformly in the argument on compact subsets of [0,1), the transient rate under two further inequalities on the parameters; and we carry every rate to the individual transition probabilities, with no monotonicity hypothesis, by a complex-analytic argument resting on a factorisation that criticality forces. The constants are made accessible by an exact integral identity for the survival probability of the system without immigration, which also yields a three-term expansion of that probability in closed form. A numerical section on explicit models confirms the constants and the rates.