Critical Markov branching process with infinite variance allowing Poisson immigration with increasing intensity (Q6637023)

The authors investigate a critical branching process with immigration \((Y(t))_{t\geq 0}\). It is assumed that (i) immigrants arrive at the epochs of an inhomogeneous Poisson process with intensity regularly varying at \(\infty\) of positive index, which is independent of the process responsible for non-immigrant particles; (ii) the numbers of immigrants arriving at different times are independent and identically distributed, and their common distribution satisfies a regular variation condition and has an infinite mean; (iii) the offspring distribution satisfies a regular variation condition and has an infinite second moment. \N\NThe authors point out three regimes depending on the interplay of the input parameters, in which the variables \(Y(t)\), properly normalized without centering, converge in distribution as \(t\to\infty\). The limit distribution is either stable (concentrated on \((0,\infty)\)) or another heavy-tailed distribution. The main technical tools of the present paper are generating functions and some results of renewal theory.