On the Structure of Entrance Laws in Discrete Spatial Critical Branching Processes
✍ Scribed by Anton Wakolbinger
- Publisher
- John Wiley and Sons
- Year
- 1991
- Tongue
- English
- Weight
- 374 KB
- Volume
- 151
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
MARKovian branching population-valued stochastic processes in discrete time are considered, in which the individuals live on a discrete space of sites and an individual a t site x produces, independently of the others, in the next generation a random offspring whose distribution depends on x, whose mean total number is assumed to be one and whose mean number a t site y is denoted by J(x, (y}). It is proved that, provided the MARKOV chain associated with the transition matrix J is null-recurrent, exactly those among the entrance laws for the population-vnlued process are extremal and have a finite mean number of individuals a t any site and time, which tire "of POISSON type", i.e. arise in a natural way from a PoIssoNian remote past. This generalizes a result of LICCETT/PORT [3] on the pure motion case to the case of branching, and also comments on a remark of DYNKIN ([I], p. 110).
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