The multitype measure branching process

1990 ◽  
Vol 22 (1) ◽  
pp. 49-67 ◽  
Author(s):  
Luis G. Gorostiza ◽  
Jose A. Lopez-Mimbela
Keyword(s):  
Small Particle ◽  
Branching Process ◽  
Martingale Problem ◽  
Single Type ◽  
System Of Particles

The existence of the multitype measure branching process is established as a small particle limit of a system of particles of several types in Rd with immigration undergoing migration, branching and mutation. The process is characterized as a solution of a martingale problem. The single-type case was studied by Dawson (1975), (1977) and Watanabe (1968).

The multitype measure branching process

1990 ◽  
Vol 22 (01) ◽  
pp. 49-67 ◽  
Author(s):  
Luis G. Gorostiza ◽  
Jose A. Lopez-Mimbela
Keyword(s):  
Small Particle ◽  
Branching Process ◽  
Martingale Problem ◽  
Single Type ◽  
System Of Particles

The existence of the multitype measure branching process is established as a small particle limit of a system of particles of several types inRdwith immigration undergoing migration, branching and mutation. The process is characterized as a solution of a martingale problem. The single-type case was studied by Dawson (1975), (1977) and Watanabe (1968).

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→∞.

Birth and death on a Brownian flow: a Feller semigroup and its generator and a martingale problem

1996 ◽  
Vol 33 (01) ◽  
pp. 71-87 ◽  
Author(s):  
Michael J. Phelan
Keyword(s):  
Markov Process ◽  
Spatial Configuration ◽  
Martingale Problem ◽  
Stochastic Flow ◽  
Transition Semigroup ◽  
Feller Semigroup ◽  
System Of Particles ◽  
Particle Process

We consider a system of particles in birth and death on a stochastic flow. The system includes a particle process tracking the spatial configuration of live particles on the flow. The particle process is a Markov process on the space of bounded counting measures. We show that its transition semigroup is a Feller semigroup and exhibit its pregenerator. The pregenerator defines a martingale problem. We show that the particle process solves the problem uniquely.

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→∞.

Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process

2004 ◽  
Vol 36 (02) ◽  
pp. 582-601 ◽  
Author(s):  
N. Lalam ◽  
C. Jacob
Keyword(s):  
Rate Of Convergence ◽  
Strong Consistency ◽  
Branching Process ◽  
Least Squares Method ◽  
Critical Size ◽  
Approximate Model ◽  
Single Type ◽  
Asymptotic Model ◽  
Transient Model ◽  
Size Dependent

We consider a single-type supercritical or near-critical size-dependent branching process {N n } n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size N n grows to ∞ and the variance may vary at the rate where −1 ≤ β < 1. The offspring mean m(N) = m + μN -α + o(N -α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {N h ,…,N n } by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model m θ,ν̂ n (·), where ν̂ n is any estimation of the parameter of the nuisance part (O(N -α) if θ0 = m and o(N -α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.

scholarly journals A Host-Parasite Model for a Two-Type Cell Population

2013 ◽  
Vol 45 (03) ◽  
pp. 719-741 ◽  
Author(s):  
Gerold Alsmeyer ◽  
Sören Gröttrup
Keyword(s):  
B Cell ◽  
Random Environment ◽  
Branching Process ◽  
Relative Proportion ◽  
Single Type ◽  
Type Model ◽  
Daughter Cells ◽  
A Cell ◽  
Host Parasite ◽  
A And B Cells

We consider a host-parasite model for a population of cells that can be of two types, A or B, and exhibits unilateral reproduction: while a B-cell always splits into two cells of the same type, the two daughter cells of an A-cell can be of any type. The random mechanism that describes how parasites within a cell multiply and are then shared into the daughter cells is allowed to depend on the hosting mother cell as well as its daughter cells. Focusing on the subpopulation of A-cells and its parasites, our model differs from the single-type model recently studied by Bansaye (2008) in that the sharing mechanism may be biased towards one of the two types. Our main results are concerned with the nonextinctive case and provide information on the behavior, as n → ∞, of the number of A-parasites in generation n and the relative proportion of A- and B-cells in this generation which host a given number of parasites. As in Bansaye (2008), proofs will make use of a so-called random cell line which, when conditioned to be of type A, behaves like a branching process in a random environment.

scholarly journals A note on the multitype measure branching process

1992 ◽  
Vol 24 (2) ◽  
pp. 496-498 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Single Type ◽  
Type Model

The existence of a class of multitype measure branching processes is deduced from a single-type model introduced by Li [8], which extends the work of Gorostiza and Lopez-Mimbela [5] and shows that the study of a multitype process can sometimes be reduced to that of a single-type one.

scholarly journals A note on the multitype measure branching process

1992 ◽  
Vol 24 (02) ◽  
pp. 496-498 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Single Type ◽  
Type Model

The existence of a class of multitype measure branching processes is deduced from a single-type model introduced by Li [8], which extends the work of Gorostiza and Lopez-Mimbela [5] and shows that the study of a multitype process can sometimes be reduced to that of a single-type one.

scholarly journals A Host-Parasite Model for a Two-Type Cell Population

2013 ◽  
Vol 45 (3) ◽  
pp. 719-741
Author(s):  
Gerold Alsmeyer ◽  
Sören Gröttrup
Keyword(s):  
B Cell ◽  
Random Environment ◽  
Branching Process ◽  
Relative Proportion ◽  
Single Type ◽  
Type Model ◽  
Daughter Cells ◽  
A Cell ◽  
Host Parasite ◽  
A And B Cells

We consider a host-parasite model for a population of cells that can be of two types, A or B, and exhibits unilateral reproduction: while a B-cell always splits into two cells of the same type, the two daughter cells of an A-cell can be of any type. The random mechanism that describes how parasites within a cell multiply and are then shared into the daughter cells is allowed to depend on the hosting mother cell as well as its daughter cells. Focusing on the subpopulation of A-cells and its parasites, our model differs from the single-type model recently studied by Bansaye (2008) in that the sharing mechanism may be biased towards one of the two types. Our main results are concerned with the nonextinctive case and provide information on the behavior, as n → ∞, of the number of A-parasites in generation n and the relative proportion of A- and B-cells in this generation which host a given number of parasites. As in Bansaye (2008), proofs will make use of a so-called random cell line which, when conditioned to be of type A, behaves like a branching process in a random environment.

Birth and death on a Brownian flow: a Feller semigroup and its generator and a martingale problem

1996 ◽  
Vol 33 (1) ◽  
pp. 71-87 ◽  
Author(s):  
Michael J. Phelan
Keyword(s):  
Markov Process ◽  
Spatial Configuration ◽  
Martingale Problem ◽  
Stochastic Flow ◽  
Transition Semigroup ◽  
Feller Semigroup ◽  
System Of Particles ◽  
Particle Process

We consider a system of particles in birth and death on a stochastic flow. The system includes a particle process tracking the spatial configuration of live particles on the flow. The particle process is a Markov process on the space of bounded counting measures. We show that its transition semigroup is a Feller semigroup and exhibit its pregenerator. The pregenerator defines a martingale problem. We show that the particle process solves the problem uniquely.

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