Decay parameter and related properties of \(n\)-type branching processes
The authors consider in this paper decay properties including the decay parameter, invariant measures, invariant vectors and quasistationary distributions for \(n\)-type Markov branching processes (\(n\)TMBPs) on the basis of the 1-type Markov branching processes and 2-type Markov branching processes. After the Introduction (Section 1), in Section 2, some deep properties of the generating functions for \(n\)-type branching (\(n\)TB) \(q\)-matrix are investigated. The properties of the generating functions \(\{B_i(u_1,\dotsc, u_n)\}\), \(i= 1,\dotsc, n\), defined at the beginning of the section are discussed. Seven lemmas are proved in this sense. In Section 3, the authors study the decay parameter for \(n\)TMBPs. Thus it is proved that, for any given \(n\)TB \(q\)-matrix \(Q\) defined in Definition 1.1, there always exists exactly one \(Q\)-process satisfying the Kolmogorov forward equation. In this sense, Theorem 3.1 proves the uniqueness property of \(n\)TMBPs; and Theorem 3.2 gives a lower bound of the decay parameter. Then the exact value of the decay parameter is presented in Theorem 3.3. In the last section, the \(\lambda_C\)-invariant vectors/measures and quasistationary distribution of \(n\)TBs are considered. In this way, Theorem 4.1 gives a \(\lambda_C\)-invariant vector for \(Q\) (or for \(P(t)\)) on \(C\); and, in Theorem 4.2, the \(\lambda_C\)-invariant measures for \(Q\) (or for \(P(t)\)) on \(C\) are considered. Then, Theorem 4.3 contains the quasistationary distributions for \(P(t)\) on \(C\). The paper is very good and instructive.
scientific article; zbMATH DE number 6131920