Birth and death processes in random environments with feedback

1987 ◽  
Vol 24 (1) ◽  
pp. 25-34 ◽  
Author(s):  
Richard Cornez
Keyword(s):  
Random Environment ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Positive Probability ◽  
Birth And Death Processes ◽  
Environmental Process ◽  
Birth And Death Chain

A generalization of a birth and death chain in a random environment (Yn, Zn) is developed allowing for feedback to the environmental process (Yn). The resulting process is then known as a birth and death chain in a random environment with feedback. Sufficient conditions are found under which the (Zn) process goes extinct almost surely or has strictly positive probability of non-extinction.

Birth and death processes in random environments with feedback

1987 ◽  
Vol 24 (01) ◽  
pp. 25-34 ◽  
Author(s):  
Richard Cornez
Keyword(s):  
Random Environment ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Positive Probability ◽  
Birth And Death Processes ◽  
Environmental Process ◽  
Birth And Death Chain

A generalization of a birth and death chain in a random environment (Y n , Z n ) is developed allowing for feedback to the environmental process (Y n ). The resulting process is then known as a birth and death chain in a random environment with feedback. Sufficient conditions are found under which the (Z n ) process goes extinct almost surely or has strictly positive probability of non-extinction.

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.

Birth and death processes with random environments in continuous time

1981 ◽  
Vol 18 (01) ◽  
pp. 19-30 ◽  
Author(s):  
Robert Cogburn ◽  
William C. Torrez
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Continuous Time ◽  
Death Process ◽  
Random Environments ◽  
Birth And Death Process ◽  
Time Model ◽  
Birth And Death Processes ◽  
Discrete Time Model ◽  
Extinction Probabilities

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

Stability of linear stochastic difference equations in strategically controlled random environments

2003 ◽  
Vol 35 (4) ◽  
pp. 961-981 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Stochastic Process ◽  
Random Environment ◽  
Linear Relation ◽  
Nash Equilibria ◽  
Stochastic Game ◽  
Sufficient Conditions ◽  
Stationary Regime ◽  
Random Environments ◽  
Simultaneous Control ◽  
Markov Strategies

We consider the stochastic sequence {Yt}t∈ℕ defined recursively by the linear relation Yt+1=AtYt+Bt in a random environment. The environment is described by the stochastic process {(At,Bt)}t∈ℕ and is under the simultaneous control of several agents playing a discounted stochastic game. We formulate sufficient conditions on the game which ensure the existence of Nash equilibria in Markov strategies which have the additional property that, in equilibrium, the process {Yt}t∈ℕ converges in distribution to a stationary regime.

Birth and death processes with random environments in continuous time

1981 ◽  
Vol 18 (1) ◽  
pp. 19-30 ◽  
Author(s):  
Robert Cogburn ◽  
William C. Torrez
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Continuous Time ◽  
Death Process ◽  
Random Environments ◽  
Birth And Death Process ◽  
Time Model ◽  
Birth And Death Processes ◽  
Discrete Time Model ◽  
Extinction Probabilities

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

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 Comparing lifetimes of coherent systems with dependent components operating in random environments

2019 ◽  
Vol 56 (3) ◽  
pp. 937-957
Author(s):  
Nil Kamal Hazra ◽  
Maxim Finkelstein
Keyword(s):  
Random Environment ◽  
Sufficient Conditions ◽  
The Other ◽  
Random Environments ◽  
Coherent System ◽  
Coherent Systems ◽  
Stochastic Comparisons ◽  
The Impact ◽  
System Operating

AbstractWe study the impact of a random environment on lifetimes of coherent systems with dependent components. There are two combined sources of this dependence. One results from the dependence of the components of the coherent system operating in a deterministic environment and the other is due to dependence of components of the system sharing the same random environment. We provide different sets of sufficient conditions for the corresponding stochastic comparisons and consider various scenarios, namely, (i) two different (as a specific case, identical) coherent systems operate in the same random environment; (ii) two coherent systems operate in two different random environments; (iii) one of the coherent systems operates in a random environment and the other in a deterministic environment. Some examples are given to illustrate the proposed reasoning.

Stability of linear stochastic difference equations in strategically controlled random environments

2003 ◽  
Vol 35 (04) ◽  
pp. 961-981 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Stochastic Process ◽  
Random Environment ◽  
Linear Relation ◽  
Nash Equilibria ◽  
Stochastic Game ◽  
Sufficient Conditions ◽  
Stationary Regime ◽  
Random Environments ◽  
Simultaneous Control ◽  
Markov Strategies

We consider the stochastic sequence {Y t } t∈ℕ defined recursively by the linear relation Y t+1=A t Y t +B t in a random environment. The environment is described by the stochastic process {(A t ,B t )} t∈ℕ and is under the simultaneous control of several agents playing a discounted stochastic game. We formulate sufficient conditions on the game which ensure the existence of Nash equilibria in Markov strategies which have the additional property that, in equilibrium, the process {Y t } t∈ℕ converges in distribution to a stationary regime.

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.

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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