Renewal, regenerative, and branching processes with stable distributions
The first part of the paper consists of a summary on earlier results of the authors. For an alternating renewal process with random waiting times following random working periods, a renewal theorem for the spent-time process is considered under the assumption that waiting and/or working periods are i.i.d.\ belonging to the domain of attraction of some stable law with index in \((\frac12,1]\). The renewal theorem is basic for the behaviour of an alternating regenerative process, where random waiting periods start whenever an evolution process hits state 0. In the second part of the paper this basic regeneration theorem is applied to branching processes of the form \[ Y_n=\Big(\sum_{i=1}^{Y_{n-1}}X_i(n)+M_n\Big)_+,\quad n\in\mathbb N, \] where \(M_n\) is additional random migration consisting of three components: family and individual emigration with probability \(p\), no migration with probability \(q\), and state-dependent immigration with probability \(r\); \(p+q+r=1\). In case \(q=1\), i.e.\ \(M_n\equiv0\) almost surely, \((Y_n)\) defines a usual Galton-Watson branching process. Particular cases of these migration components have been studied earlier. Since state 0 is a reflective barrier for the process \((Y_n)\), it can also be seen as an alternating regenerative process, allowing to consider all the migration components at the same time.
scientific article; zbMATH DE number 2183291