Coalescence on critical and subcritical multitype branching processes

2016 ◽  
Vol 53 (3) ◽  
pp. 802-817
Author(s):  
Jyy-I Hong
Keyword(s):  
Limit Distribution ◽  
Joint Distribution ◽  
Branching Process ◽  
Common Ancestor ◽  
Branching Processes ◽  
Recent Common Ancestor ◽  
Most Recent Common Ancestor ◽  
Generation Number ◽  
Multitype Branching Processes ◽  
Lines Of Descent

AbstractConsider a d-type (d<∞) Galton–Watson branching process, conditioned on the event that there are at least k≥2 individuals in the nth generation, pick k individuals at random from the nth generation and trace their lines of descent backward in time till they meet. In this paper, the limit behaviors of the distributions of the generation number of the most recent common ancestor of any k chosen individuals and of the whole population are studied for both critical and subcritical cases. Also, we investigate the limit distribution of the joint distribution of the generation number and their types.

The genealogy of branching processes and the age of our most recent common ancestor

1995 ◽  
Vol 27 (02) ◽  
pp. 418-442 ◽  
Author(s):  
Neil O'Connell
Keyword(s):  
Dna Sequences ◽  
Branching Process ◽  
Common Ancestor ◽  
Branching Processes ◽  
Recent Common Ancestor ◽  
Time Interval ◽  
Weak Approximation ◽  
Family Tree ◽  
Exponential Law ◽  
Most Recent Common Ancestor

We obtain a weak approximation for the reduced family tree in a near-critical Markov branching process when the time interval considered is long; we also extend Yaglom's theorem and the exponential law to this case. These results are then applied to the problem of estimating the age of our most recent common female ancestor, using mitochondrial DNA sequences taken from a sample of contemporary humans.

The genealogy of branching processes and the age of our most recent common ancestor

1995 ◽  
Vol 27 (2) ◽  
pp. 418-442 ◽  
Author(s):  
Neil O'Connell
Keyword(s):  
Dna Sequences ◽  
Branching Process ◽  
Common Ancestor ◽  
Branching Processes ◽  
Recent Common Ancestor ◽  
Time Interval ◽  
Weak Approximation ◽  
Family Tree ◽  
Exponential Law ◽  
Most Recent Common Ancestor

We obtain a weak approximation for the reduced family tree in a near-critical Markov branching process when the time interval considered is long; we also extend Yaglom's theorem and the exponential law to this case. These results are then applied to the problem of estimating the age of our most recent common female ancestor, using mitochondrial DNA sequences taken from a sample of contemporary humans.

scholarly journals Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

2013 ◽  
Vol 50 (2) ◽  
pp. 576-591
Author(s):  
Jyy-I Hong
Keyword(s):  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Simple Random Sampling ◽  
Single Type ◽  
Last Common Ancestor ◽  
Death Time ◽  
Age Dependent ◽  
Lines Of Descent ◽  
First Time

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={at,i: 1≤ i≤ Z(t)}, where at,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of Dk(t) and its limit distribution as t→∞.

Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

2013 ◽  
Vol 50 (02) ◽  
pp. 576-591 ◽  
Author(s):  
Jyy-I Hong
Keyword(s):  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Simple Random Sampling ◽  
Single Type ◽  
Last Common Ancestor ◽  
Death Time ◽  
Age Dependent ◽  
Lines Of Descent ◽  
First Time

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={a t,i : 1≤ i≤ Z(t)}, where a t,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let D k(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of D k(t) and its limit distribution as t→∞.

Coalescence Times for the Bienaymé-Galton-Watson Process

2014 ◽  
Vol 51 (01) ◽  
pp. 209-218
Author(s):  
V. Le
Keyword(s):  
Continuous Time ◽  
Joint Distribution ◽  
Common Ancestor ◽  
Recent Common Ancestor ◽  
Subcritical Case ◽  
Current Generation ◽  
Most Recent Common Ancestor ◽  
Watson Process ◽  
Number Of Individuals ◽  
Coalescence Times

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a continuous-time Bienaymé-Galton-Watson process foundedtunits of time ago. We also obtain limiting distributions ast→ ∞ in the subcritical case. We extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

scholarly journals Coalescence Times for the Bienaymé-Galton-Watson Process

2014 ◽  
Vol 51 (1) ◽  
pp. 209-218 ◽  
Author(s):  
V. Le
Keyword(s):  
Continuous Time ◽  
Joint Distribution ◽  
Common Ancestor ◽  
Recent Common Ancestor ◽  
Subcritical Case ◽  
Current Generation ◽  
Most Recent Common Ancestor ◽  
Watson Process ◽  
Number Of Individuals ◽  
Coalescence Times

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a continuous-time Bienaymé-Galton-Watson process founded t units of time ago. We also obtain limiting distributions as t → ∞ in the subcritical case. We extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process

1989 ◽  
Vol 26 (01) ◽  
pp. 1-8
Author(s):  
V. G. Gadag ◽  
M. B. Rajarshi
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
White Male ◽  
Asymptotic Results ◽  
Male Population ◽  
Multitype Branching Process ◽  
Original Process ◽  
The Usa ◽  
Multitype Branching Processes ◽  
Lines Of Descent

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.

Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process

1989 ◽  
Vol 26 (1) ◽  
pp. 1-8 ◽  
Author(s):  
V. G. Gadag ◽  
M. B. Rajarshi
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
White Male ◽  
Asymptotic Results ◽  
Male Population ◽  
Multitype Branching Process ◽  
Original Process ◽  
The Usa ◽  
Multitype Branching Processes ◽  
Lines Of Descent

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.

scholarly journals Skeletal stochastic differential equations for superprocesses

2020 ◽  
Vol 57 (4) ◽  
pp. 1111-1134
Author(s):  
Dorottya Fekete ◽  
Joaquin Fontbona ◽  
Andreas E. Kyprianou
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Subcritical Case ◽  
Markov Branching Process ◽  
Continuous State ◽  
Common Framework ◽  
Infinite Line ◽  
Current Article ◽  
Lines Of Descent ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

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