Limit distributions for branching processes with immigration
Let \(f(z)\) be the infinitesimal generating function of the number of children of a particle of the basic process and \(g(z)\) be the infinitesimal generating function of the number of immigrants in a Markov continuous time branching process \(\mu (t)\) with immigration. It is know that if \(f'(1)<0\) and \(g'(1)<\infty,\) then \(\lim_{t\to\infty}P\{\mu (t) =k\}=P_k\) exists for any \(k=0,1,\ldots,\) and \(\sum_{k=0}^{\infty} P_k=1.\) The author solves, in a sense, an inverse problem. Namely, she shows that, for a fixed \(g(z),\) a probability distribution \(\{P_k\), \(k=0,1,\ldots\}\) may be limiting for a Markov continuous time branching process with immigration if and only if \(P_k\) obey a system of inequalities expressed in terms of the coefficients in the Taylor expansion of \(g(z).\)
scientific article; zbMATH DE number 1541564
scientific article