Diffusion approximation of branching migration processes
Consider a branching process \(\{Z^{(k)}_0,Z^{(k)}_1,\ldots\}\) with \(Z^{(k)}_0=k\), such that for \(Z^{(k)}_n>0\), either \(j\geq 0\) particles immigrate into the population \(Z^{(k)}_n\) with probability \(q_j\) or one particle emigrates with probability \(r\) (\(\sum_{j=0}^\infty q_j+r=1\)). After this, all particles independently produce offspring according to the same probability distribution, the total number of offspring being \(Z^{(k)}_{n+1}\). Assume that the offspring mean is \(1+\alpha/k\) and the offspring second factorial moment is \(2b\). It is shown that subject to regularity conditions, for any \(t\geq 0\), \(Z_{[kt]}^{(k)}/k\) converges in distribution as \(k\to\infty\) to \(x_t\), where \(\{x_t, t\geq 0\}\), \(x_0=1\), is the diffusion with infinitesimal drift \(\alpha x\) and variance \(bx\), having zero as an absorbing barrier.
scientific article; zbMATH DE number 1295565