On multitype branching processes in a random environment

1981 ◽  
Vol 13 (3) ◽  
pp. 464-497 ◽  
Author(s):  
David Tanny
Keyword(s):  
Random Environment ◽  
Stability Condition ◽  
Branching Processes ◽  
Classification Theorem ◽  
Regularity Conditions ◽  
Exponential Rate ◽  
Probabilistic Computation ◽  
Multitype Branching Processes ◽  
The Stability ◽  
Functional Iteration

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.

On multitype branching processes in a random environment

1981 ◽  
Vol 13 (03) ◽  
pp. 464-497 ◽  
Author(s):  
David Tanny
Keyword(s):  
Random Environment ◽  
Stability Condition ◽  
Branching Processes ◽  
Classification Theorem ◽  
Regularity Conditions ◽  
Exponential Rate ◽  
Probabilistic Computation ◽  
Multitype Branching Processes ◽  
The Stability ◽  
Functional Iteration

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.

scholarly journals Multitype branching processes in random environments

1971 ◽  
Vol 8 (1) ◽  
pp. 17-31 ◽  
Author(s):  
Edward W. Weissner
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Restrictive Assumption ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

Multitype branching processes in random environment

2021 ◽  
Vol 76 (6) ◽  
Author(s):  
Vladimir Alekseevich Vatutin ◽  
Elena Evgen'evna D'yakonova
Keyword(s):  
Random Environment ◽  
Branching Processes ◽  
Multitype Branching Processes

scholarly journals Multitype branching processes in random environments

1971 ◽  
Vol 8 (01) ◽  
pp. 17-31 ◽  
Author(s):  
Edward W. Weissner
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Restrictive Assumption ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

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