On some classes of population-size-dependent Galton–Watson processes

1985 ◽  
Vol 22 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Reinhard Höpfner
Keyword(s):  
Population Size ◽  
Lower Bounds ◽  
Generating Functions ◽  
Transition Probability ◽  
Limit Distributions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Iteration Methods ◽  
Functional Iteration

Some classes of population-size-dependent Galton–Watson processes {Z(t)}t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.

On some classes of population-size-dependent Galton–Watson processes

1985 ◽  
Vol 22 (01) ◽  
pp. 25-36
Author(s):  
Reinhard Höpfner
Keyword(s):  
Population Size ◽  
Lower Bounds ◽  
Generating Functions ◽  
Transition Probability ◽  
Limit Distributions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Iteration Methods ◽  
Functional Iteration

Some classes of population-size-dependent Galton–Watson processes {Z(t)} t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.

Bisexual Galton-Watson branching process with population-size-dependent mating

2002 ◽  
Vol 39 (3) ◽  
pp. 479-490 ◽  
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Population Size ◽  
Generating Functions ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Necessary And Sufficient ◽  
Mating Function

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.

Bisexual Galton-Watson branching process with population-size-dependent mating

2002 ◽  
Vol 39 (03) ◽  
pp. 479-490 ◽  
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Population Size ◽  
Generating Functions ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Necessary And Sufficient ◽  
Mating Function

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.

Some results on population-size-dependent Galton-Watson processes

1986 ◽  
Vol 23 (02) ◽  
pp. 297-306 ◽  
Author(s):  
Reinhard Höpfner
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Limit Distributions ◽  
Probability Of Survival ◽  
Size Dependent

Some classes of population-size-dependent Galton-Watson processes are considered where extinction occurs with probability 1. Results on the asymptotic behaviour of the probability of survival up to time t, mean population size and conditioned limit distributions are found to hold. They correspond to those obtained in the study of Galton-Watson processes with immigration stopped at 0.

A random walk problem with correlation

1972 ◽  
Vol 9 (02) ◽  
pp. 436-440 ◽  
Author(s):  
A. D. Proudfoot ◽  
D. G. Lampard
Keyword(s):  
Random Walk ◽  
Green’S Function ◽  
Green's Function ◽  
Generating Functions ◽  
Transition Probability ◽  
Higher Order ◽  
Order Transition ◽  
Probability Generating Functions ◽  
Discrete Domain

The higher-order transition probability generating functions for a random-walk with correlation between steps is calculated as a discrete-domain Green's function.

Some results on population-size-dependent Galton-Watson processes

1986 ◽  
Vol 23 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Reinhard Höpfner
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Limit Distributions ◽  
Probability Of Survival ◽  
Size Dependent

Some classes of population-size-dependent Galton-Watson processes are considered where extinction occurs with probability 1. Results on the asymptotic behaviour of the probability of survival up to time t, mean population size and conditioned limit distributions are found to hold. They correspond to those obtained in the study of Galton-Watson processes with immigration stopped at 0.

Finite dams with inputs forming a Markov chain

1970 ◽  
Vol 7 (02) ◽  
pp. 291-303 ◽  
Author(s):  
M.S. Ali Khan
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Generating Functions ◽  
State Transition ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Time Dependent ◽  
Probability Generating Functions ◽  
State Transition Probability ◽  
Finite Dam

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.

Finite dams with inputs forming a Markov chain

1970 ◽  
Vol 7 (2) ◽  
pp. 291-303 ◽  
Author(s):  
M.S. Ali Khan
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Generating Functions ◽  
State Transition ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Time Dependent ◽  
Probability Generating Functions ◽  
State Transition Probability ◽  
Finite Dam

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.

A random walk problem with correlation

1972 ◽  
Vol 9 (2) ◽  
pp. 436-440 ◽  
Author(s):  
A. D. Proudfoot ◽  
D. G. Lampard
Keyword(s):  
Random Walk ◽  
Green’S Function ◽  
Green's Function ◽  
Generating Functions ◽  
Transition Probability ◽  
Higher Order ◽  
Order Transition ◽  
Probability Generating Functions ◽  
Discrete Domain

The higher-order transition probability generating functions for a random-walk with correlation between steps is calculated as a discrete-domain Green's function.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

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