The latent roots of certain Markov chains arising in genetics: A new approach, I. Haploid models

1974 ◽  
Vol 6 (2) ◽  
pp. 260-290 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chains ◽  
Genetic Drift ◽  
Overlapping Generations ◽  
Branching Processes ◽  
Multiple Alleles ◽  
New Approach ◽  
Constant Size ◽  
Latent Roots

Haploid models of genetic drift in populations of constant size are considered. Generalizations of the models of Moran and Wright have been developed by Karlin and McGregor (for multiple alleles and non-overlapping generations), by Chia and Watterson (for two alleles and overlapping or non-overlapping generations) and by Chia (for multiple alleles and overlapping or non-overlapping generations), using conditioned branching processes. A new approach is developed which contains the models mentioned above and provides simpler expressions for the latent roots. A greater dependence between the birth events and death events can be permitted, and non-independent mutations treated.

The latent roots of certain Markov chains arising in genetics: A new approach, I. Haploid models

1974 ◽  
Vol 6 (02) ◽  
pp. 260-290 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chains ◽  
Genetic Drift ◽  
Overlapping Generations ◽  
Branching Processes ◽  
Multiple Alleles ◽  
New Approach ◽  
Constant Size ◽  
Latent Roots

Haploid models of genetic drift in populations of constant size are considered. Generalizations of the models of Moran and Wright have been developed by Karlin and McGregor (for multiple alleles and non-overlapping generations), by Chia and Watterson (for two alleles and overlapping or non-overlapping generations) and by Chia (for multiple alleles and overlapping or non-overlapping generations), using conditioned branching processes. A new approach is developed which contains the models mentioned above and provides simpler expressions for the latent roots. A greater dependence between the birth events and death events can be permitted, and non-independent mutations treated.

The latent roots of certain Markov chains arising in genetics: A new approach, II. Further haploid models

1975 ◽  
Vol 7 (02) ◽  
pp. 264-282 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Reproductive Structure ◽  
Variable Size ◽  
New Approach ◽  
Age Structured ◽  
Drift Models ◽  
Latent Roots

The method developed for the treatment of the classical drift models of Wright and Moran, and their generalizations, in Cannings (1974) are extended to more complex haploid models. The possibility of subdivision of the population, as for migration models and age-structured models, is incorporated. Models with variable size or reproductive structure determined by another Markov chain are analysed.

The latent roots of certain Markov chains arising in genetics: A new approach, II. Further haploid models

1975 ◽  
Vol 7 (2) ◽  
pp. 264-282 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Reproductive Structure ◽  
Variable Size ◽  
New Approach ◽  
Age Structured ◽  
Drift Models ◽  
Latent Roots

The method developed for the treatment of the classical drift models of Wright and Moran, and their generalizations, in Cannings (1974) are extended to more complex haploid models. The possibility of subdivision of the population, as for migration models and age-structured models, is incorporated. Models with variable size or reproductive structure determined by another Markov chain are analysed.

Near-constancy phenomena in branching processes

1991 ◽  
Vol 110 (3) ◽  
pp. 545-558 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Recent Work ◽  
Branching Processes ◽  
New Approach ◽  
Primary Motivation

The occurrence of certain ‘near-constancy phenomena’ in some aspects of the theory of (simple) branching processes forms the background for the work below. The problem arises out of work by Karlin and McGregor [8, 9]. A detailed study of the theoretical and numerical aspects of the Karlin–McGregor near-constancy phenomenon was given by Dubuc[7], and considered further by Bingham[4]. We give a new approach which simplifies and generalizes the results of these authors. The primary motivation for doing this was the recent work of Barlow and Perkins [3], who observed near-constancy in a framework not immediately covered by the results then known.

Branching processes, Markov chains, and competing risks

1978 ◽  
Vol 1 (6) ◽  
pp. 341-343
Author(s):  
Wolfgang J. Bühler
Keyword(s):  
Markov Chains ◽  
Competing Risks ◽  
Branching Processes

scholarly journals On sojourn times at particular gene frequencies

1975 ◽  
Vol 25 (2) ◽  
pp. 89-94 ◽  
Author(s):  
Edward Pollak ◽  
Barry C. Arnold
Keyword(s):  
Markov Chains ◽  
Gene Frequency ◽  
Overlapping Generations ◽  
Finite Population ◽  
Diffusion Method ◽  
Gene Frequencies ◽  
Sojourn Times ◽  
The Mean ◽  
Number Of Visits ◽  
Finite Markov Chains

SUMMARYThe distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of bij's, where bij is the mean number of visits to frequency j/2N, given initial frequency i/2N. The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.

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.

Approximating Transition Probabilities and Mean Occupation Times in Continuous-Time Markov Chains

1987 ◽  
Vol 1 (3) ◽  
pp. 251-264 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Random Time ◽  
Occupation Times ◽  
New Approach ◽  
Continuous Time Markov Chains ◽  
The Mean ◽  
Mean Time

In this paper we propose a new approach for estimating the transition probabilities and mean occupation times of continuous-time Markov chains. Our approach is to approximate the probability of being in a state (or the mean time already spent in a state) at time t by the probability of being in that state (or the mean time already spent in that state) at a random time that is gamma distributed with mean t.

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.

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