An infinite-alleles version of the simple branching process

1988 ◽  
Vol 20 (3) ◽  
pp. 489-524 ◽  
Author(s):  
R. C. Griffiths ◽  
Anthony G. Pakes
Keyword(s):  
Frequency Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Expected Number ◽  
Generation Number ◽  
Tail Behaviour ◽  
Mutation Event

Individuals in a population which grows according to the rules defining the simple branching process can mutate to novel allelic forms. We obtain limit theorems for the number of alleles present in any generation, the total number of alleles ever seen and the number of the generation containing the last mutation event.In addition we define a notion of frequency spectrum for each generation as the expected number of alleles having a given number of representatives. As the generation number increases we prove the existence of a limiting notion of the frequency spectrum and discuss its upper tail behaviour. Our results here are incomplete and we make some conjectures which are supported by informal argument and specific examples.

An infinite-alleles version of the simple branching process

1988 ◽  
Vol 20 (03) ◽  
pp. 489-524 ◽  
Author(s):  
R. C. Griffiths ◽  
Anthony G. Pakes
Keyword(s):  
Frequency Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Expected Number ◽  
Generation Number ◽  
Tail Behaviour ◽  
Mutation Event

Individuals in a population which grows according to the rules defining the simple branching process can mutate to novel allelic forms. We obtain limit theorems for the number of alleles present in any generation, the total number of alleles ever seen and the number of the generation containing the last mutation event.In addition we define a notion of frequency spectrum for each generation as the expected number of alleles having a given number of representatives. As the generation number increases we prove the existence of a limiting notion of the frequency spectrum and discuss its upper tail behaviour. Our results here are incomplete and we make some conjectures which are supported by informal argument and specific examples.

scholarly journals A multitype infinite-allele branching process with applications to cancer evolution

2015 ◽  
Vol 52 (03) ◽  
pp. 864-876 ◽  
Author(s):  
Thomas O. McDonald ◽  
Marek Kimmel
Keyword(s):  
Frequency Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Cancer Evolution

We extend the infinite-allele simple branching process of Griffiths and Pakes (1988) allowing the offspring to change types and labels. The model is developed and limit theorems are given for the growth of the number of labels of a specific type. We also discuss the asymptotics of the frequency spectrum. Finally, we present an application of the model's use in tumorigenesis.

scholarly journals A multitype infinite-allele branching process with applications to cancer evolution

2015 ◽  
Vol 52 (3) ◽  
pp. 864-876 ◽  
Author(s):  
Thomas O. McDonald ◽  
Marek Kimmel
Keyword(s):  
Frequency Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Cancer Evolution

We extend the infinite-allele simple branching process of Griffiths and Pakes (1988) allowing the offspring to change types and labels. The model is developed and limit theorems are given for the growth of the number of labels of a specific type. We also discuss the asymptotics of the frequency spectrum. Finally, we present an application of the model's use in tumorigenesis.

Some properties of a branching process with group immigration and emigration

1986 ◽  
Vol 18 (3) ◽  
pp. 628-645 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Renewal Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Limiting Distribution ◽  
Immigration And Emigration ◽  
Event Times

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.

Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (04) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

Limit theorems for processes such as the Markov branching process

1975 ◽  
Vol 12 (02) ◽  
pp. 289-297
Author(s):  
Andrew D. Barbour
Keyword(s):  
Markov Process ◽  
Residence Time ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Continuous Mapping ◽  
Random Variables ◽  
Time Parameter ◽  
Image Position ◽  
Independent Identically Distributed

LetX(t) be a continuous time Markov process on the integers such that, ifσis a time at whichXmakes a jump,X(σ)– X(σ–) is distributed independently ofX(σ–), and has finite meanμand variance. Letq(j) denote the residence time parameter for the statej.Iftndenotes the time of thenth jump andXn≡X(tb), it is easy to deduce limit theorems forfrom those for sums of independent identically distributed random variables. In this paper, it is shown how, forμ> 0 and for suitableq(·), these theorems can be translated into limit theorems forX(t), by using the continuous mapping theorem.

The expected population size in a cell-size-dependent branching process

1981 ◽  
Vol 18 (01) ◽  
pp. 65-75 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Cell Size ◽  
Exponential Growth ◽  
Branching Process ◽  
Expected Number ◽  
Size Dependent ◽  
Daughter Cells ◽  
Rate Of Increase ◽  
A Cell ◽  
Dependent Growth ◽  
Number Of Cells

In cell-size-dependent growth the probabilistic rate of division of a cell into daughter-cells and the rate of increase of its size depend on its size. In this paper the expected number of cells in the population at time t is calculated for a variety of models, and it is shown that population growths slower and faster than exponential are both possible. When the cell sizes are bounded conditions are given for exponential growth.

Some limit theorems for the total progeny of a branching process

1971 ◽  
Vol 3 (1) ◽  
pp. 176-192 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Total Progeny

We consider a branching process in which each individual reproduces independently of all others and has probability aj(j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 < a0, a0 + a1 < 1.

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