scholarly journals Subcritical branching processes in a random environment without the Cramer condition

2012 ◽  
Vol 122 (7) ◽  
pp. 2594-2609 ◽  
Author(s):  
Vladimir Vatutin ◽  
Xinghua Zheng
Keyword(s):  
Random Environment ◽  
Branching Processes ◽  
Cramer Condition ◽  
Cramér Condition

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.

scholarly journals Two-Sex Branching Processes with Offspring and Mating in a Random Environment

2009 ◽  
Vol 46 (04) ◽  
pp. 993-1004
Author(s):  
S. Ma ◽  
M. Molina
Keyword(s):  
Mathematical Model ◽  
Probability Distribution ◽  
Discrete Time ◽  
Random Environment ◽  
Generating Functions ◽  
Branching Processes ◽  
Random Vectors ◽  
Environmental Process ◽  
Mating Function ◽  
The Mathematical Model

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.

Random environment branching processes with equal environmental extinction probabilities

1973 ◽  
Vol 10 (03) ◽  
pp. 659-665
Author(s):  
Donald C. Raffety
Keyword(s):  
Markov Chains ◽  
Population Size ◽  
Random Environment ◽  
Transition Matrix ◽  
Branching Processes ◽  
Random Variables ◽  
Conditional Probabilities ◽  
Left Eigenvector ◽  
Extinction Probabilities ◽  
Limiting Values

R-positivity theory for Markov chains is used to obtain results for random environment branching processes whose environment random variables are independent and identically distributed and whose environmental extinction probabilities are equal. For certain processes whose eventual extinction is almost sure, it is shown that the distribution of population size conditioned by non-extinction at time n tends to a left eigenvector of the transition matrix. Limiting values of other conditional probabilities are given in terms of this left eigenvector and it is shown that the probability of non-extinction at time n approaches zero geometrically as n approaches ∞. Analogous results are obtained for processes whose extinction is not almost sure.

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