期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2018
卷号：XV
期号：1
页码：409-428
DOI：10.30757/ALEA.v15-17
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：In Li (2011, Example 2.2), the notion of a multi-type continuous-statebranching process (MCSBP) was introduced with a finite number of types, withthe countably infinite case being proposed in Kyprianou and Palau (2018+). Onemay consider such processes as a super-Markov chain on a countable state-spaceof types, which undertakes both local and non-local branching. In Kyprianou andPalau (2018+) it was shown that, for MCSBPs, under mild conditions, there existsa lead eigenvalue which characterises the spectral radius of the linear semigroupassociated to the process. Moreover, in a qualitative sense, the sign of this eigen-value distinguishes between the cases where there is local extinction and exponentialgrowth. In this paper, we continue in this vein and show that, when the numberof types is finite, the lead eigenvalue gives the precise almost sure rate of growth ofeach type. This result matches perfectly classical analogues for multi-type Galton–Watson processes.
关键词：Continuous;state branching processes; non;local branching mechanism;super Markov chain; strong law of large numbers
