Branching processes with dependence but homogeneous growth
Let \(\tau_x\) be the birth time of an individual \(x\). Denote the number of all individuals alive at \(t\) by \(Z_t\) (general branching process with possible dependence between individuals) and denote the number of individuals stemming from \(x\) and alive at \(t\) by \(Z_t(x)\). Let \(\mathcal G_x\) be the \(\sigma\)-algebra containing all information about matters that occurred up to \(\tau_x\). The process \(Z_t\) is said to grow homogeneously (or to exhibit symmetric growth) if there is a constant \(K\) such that \[ \mathbf E \left[\frac{Z_{t-\tau_x}(x)}{Z_t} \biggl|\mathcal G_x\right]\leq \frac{K}{Z_{\tau_x}} \] for all individuals \(x\) and \(t\geq \tau_x\). All individuals in \(Z_n\) are numbered in some order and \(\tau_i\), \(i\in N\), are their birth times. It is defined a general supercritical Malthusian branching population with minorizing reproduction. These two processes can be coupled through imaginary abortions and \(\delta_i\) indicate abortion or not for the particle number \(i\) in the additional process. The main result is: if \(W_t:=\exp (-\alpha t)Z_t\) is tight, then \(\sum_i \mathbf E[\delta_i/Z_{\tau_i}]<\infty\) implies that \(W_t\) tends in probability to some finite \(W\geq 0\), not identically zero on the set of nonextinction. This is a continuation of the author's article [Ann. Appl. Probab. 7, No. 2, 281-298 (1997; Zbl 0884.60084)]. The next purpose of the paper is to establish Klebaner-type criteria [see \textit{F. C. Klebaner}, J. Appl. Probab. 21, 40-49 (1984; Zbl 0544.60073), ibid. 22, 48-57 (1985; Zbl 0562.60092) and Ann. Probab. 17, No. 4, 1466-1477 (1989; Zbl 0694.60079)] for simple and certain age-dependent branching processes as a corollary of the main result.
scientific article; zbMATH DE number 1500268