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.

Random environment branching processes with equal environmental extinction probabilities

1973 ◽  
Vol 10 (3) ◽  
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.

On determining absorption probabilities for Markov chains in random environments

1981 ◽  
Vol 13 (2) ◽  
pp. 369-387 ◽  
Author(s):  
Richard D. Bourgin ◽  
Robert Cogburn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Environment ◽  
Efficient Method ◽  
General Framework ◽  
Random Environments ◽  
Extinction Probabilities ◽  
Environmental Process ◽  
Branching Chains ◽  
Absorption Probabilities

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.

On the fluctuation of stochastically monotone Markov chains and some applications

1983 ◽  
Vol 20 (1) ◽  
pp. 178-184 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chains ◽  
Transition Probabilities ◽  
Branching Processes ◽  
Random Variables ◽  
One Step ◽  
Type Property ◽  
Step Transition ◽  
Varying Environment

A Borel–Cantelli-type property in terms of one-step transition probabilities is given for events like {|Xn+1| > a + ε, |Xn|≦a}, a and ε being two positive numbers. Applications to normed sums of i.i.d. random variables with infinite mean and branching processes in varying environment with or without immigration are derived.

Large deviations of branching process in a random environment

2021 ◽  
Vol 31 (4) ◽  
pp. 281-291
Author(s):  
Aleksandr V. Shklyaev
Keyword(s):  
Difference Equation ◽  
Large Deviations ◽  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Asymptotic Formulas ◽  
Random Difference Equation ◽  
Random Difference ◽  
Independent Identically Distributed

Abstract In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Y n+1=A n Y n + B n , where A 1, A 2, … are independent identically distributed random variables and B n may depend on { ( A k , B k ) , 0 ⩽ k < n } $ \{(A_k,B_k),0\leqslant k \lt n\} $ for any n≥1. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.

Occupation times for two state Markov chains

1971 ◽  
Vol 8 (02) ◽  
pp. 381-390 ◽  
Author(s):  
P. J. Pedler
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Random Variables ◽  
Time Parameter ◽  
Conditional Probabilities ◽  
Occupation Times ◽  
The Matrix ◽  
Image Position

Consider first a Markov chain with two ergodic states E 1 and E 2, and discrete time parameter set {0, 1, 2, ···, n}. Define the random variables Z 0, Z 1, Z 2, ···, Zn by then the conditional probabilities for k = 1,2,···, n, are independent of k. Thus the matrix of transition probabilities is

On the fluctuation of stochastically monotone Markov chains and some applications

1983 ◽  
Vol 20 (01) ◽  
pp. 178-184 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chains ◽  
Transition Probabilities ◽  
Branching Processes ◽  
Random Variables ◽  
One Step ◽  
Type Property ◽  
Step Transition ◽  
Varying Environment

A Borel–Cantelli-type property in terms of one-step transition probabilities is given for events like {|Xn +1| &gt; a + ε, |Xn|≦a}, a and ε being two positive numbers. Applications to normed sums of i.i.d. random variables with infinite mean and branching processes in varying environment with or without immigration are derived.

Occupation times for two state Markov chains

1971 ◽  
Vol 8 (2) ◽  
pp. 381-390 ◽  
Author(s):  
P. J. Pedler
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Random Variables ◽  
Time Parameter ◽  
Conditional Probabilities ◽  
Occupation Times ◽  
The Matrix ◽  
Image Position

Consider first a Markov chain with two ergodic states E1 and E2, and discrete time parameter set {0, 1, 2, ···, n}. Define the random variables Z0, Z1, Z2, ···, Znby then the conditional probabilities for k = 1,2,···, n, are independent of k. Thus the matrix of transition probabilities is

On determining absorption probabilities for Markov chains in random environments

1981 ◽  
Vol 13 (02) ◽  
pp. 369-387 ◽  
Author(s):  
Richard D. Bourgin ◽  
Robert Cogburn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Environment ◽  
Efficient Method ◽  
General Framework ◽  
Random Environments ◽  
Extinction Probabilities ◽  
Environmental Process ◽  
Branching Chains ◽  
Absorption Probabilities

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.

scholarly journals On the extinction times of varying and random environment branching processes

1975 ◽  
Vol 12 (01) ◽  
pp. 39-46 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Extinction Time ◽  
Probability Of Extinction ◽  
Stochastically Bounded ◽  
Necessary And Sufficient ◽  
Varying Environment

Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.

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