The asymptotic behavior of the Perron root for a family of branching processes. (English. Ukrainian original) Zbl 0948.60078

Jelejko, Ya. I.

Theory Probab. Math. Stat. 52, 69-74 (1996); translation from Teor. Jmovirn. Mat. Stat. 52, 66-71 (1995).

Consider a family of Bellman-Harris branching processes with particles of \(n\) types. Let \(x_j^{\varepsilon}(t)\) be the number of particles of the \(n\)th type at the moment \(t.\) Put \(K_{\varepsilon}=\|E_i x_j^{\varepsilon}(\tau)\|,\) where \(\tau\) is the moment of the first reproduction and \(E_i\) denotes the mathematical expectation provided \(x_j^{\varepsilon}(0)=\delta_j^i.\) Assume that \(K_{\varepsilon}\to K\) as \(\varepsilon\to 0,\) where \(K\) is a decomposable matrix with the maximal eigenvalue 1. In this case \(K_{\varepsilon}\) has the Perron root \(\rho_{\varepsilon},\) and \(\rho_{\varepsilon}\to 1\) as \(\varepsilon\to 0.\) The paper describes the asymptotic behavior of \(\rho_{\varepsilon}-1\) as \(\varepsilon\to 0.\)

Reviewer: O.K.Zakusilo (Kyïv)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J85	Applications of branching processes
92D25	Population dynamics (general)

Keywords:

Perron root; branching process; Bellman-Harris process