Conditional moments in a critical age-dependent branching process

1975 ◽  
Vol 12 (3) ◽  
pp. 581-587 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Asymptotic Behavior ◽  
Branching Process ◽  
Conditional Moments ◽  
Age Dependent ◽  
Critical Age ◽  
Number Of Cells

Let X(t), N(t) respectively denote the number of cells alive at t and the total number of cells born by t in a critical age-dependent Bellman-Harris branching process.The asymptotic behavior of the conditional moments, for 0 < α ≦ 1, E(Nn(αt) | X(t) > 0), E(Nn(t) |X(αt) > 0), is obtained.

Conditional moments in a critical age-dependent branching process

1975 ◽  
Vol 12 (03) ◽  
pp. 581-587
Author(s):  
Howard J. Weiner
Keyword(s):  
Asymptotic Behavior ◽  
Branching Process ◽  
Conditional Moments ◽  
Age Dependent ◽  
Critical Age ◽  
Number Of Cells

Let X(t), N(t) respectively denote the number of cells alive at t and the total number of cells born by t in a critical age-dependent Bellman-Harris branching process. The asymptotic behavior of the conditional moments, for 0 &lt; α ≦ 1, E(Nn (αt) | X(t) &gt; 0), E(Nn (t) |X(αt) &gt; 0), is obtained.

Total progeny in a multitype critical age dependent branching process with immigration

1974 ◽  
Vol 11 (3) ◽  
pp. 458-470 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Cell Types ◽  
Dimensional Case ◽  
One Dimensional ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The One ◽  
Branching Process With Immigration

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

Asymptotic probabilities in a critical age-dependent branching process

1976 ◽  
Vol 13 (3) ◽  
pp. 476-485 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Branching Process ◽  
Comparison Method ◽  
Lifetime Distribution ◽  
Order Moment ◽  
Moment Condition ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The Mean ◽  
Critical Process

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.The method is also applied to the total progeny in the critical process.

On a multi-type critical age-dependent branching process

1970 ◽  
Vol 7 (3) ◽  
pp. 523-543 ◽  
Author(s):  
H. J. Weiner
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Continuous Distribution ◽  
Cell Type ◽  
Distinguishable Particle ◽  
Age Dependent ◽  
Critical Age ◽  
Image Position

We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi(t), Gi(0+) = 0. At the end of its life, cell i is replaced by j1 new cells of type 1, j2 new cells of type 2, ···, jm new cells of type m with probability , and we define the generating functions for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).

On a multi-type critical age-dependent branching process

1970 ◽  
Vol 7 (03) ◽  
pp. 523-543 ◽  
Author(s):  
H. J. Weiner
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Continuous Distribution ◽  
Cell Type ◽  
Distinguishable Particle ◽  
Age Dependent ◽  
Critical Age ◽  
Image Position

We will consider a branching process with m &gt; 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi (t), Gi (0+) = 0. At the end of its life, cell i is replaced by j 1 new cells of type 1, j 2 new cells of type 2, ···, jm new cells of type m with probability , and we define the generating functions for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).

A multi-type critical age-dependent branching process with immigration

1972 ◽  
Vol 9 (4) ◽  
pp. 697-706 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Gamma Distribution ◽  
Generating Functions ◽  
Branching Process ◽  
Cell Types ◽  
Distribution Law ◽  
One Dimensional ◽  
Age Dependent ◽  
Critical Age ◽  
Branching Process With Immigration

In a multi-type critical age-dependent branching process with immigration, the numbers of cell types alive at time t, each divided by t, as t becomes large, tends to a one-dimensional gamma distribution law. The method of proof employs generating functions and compution of asymptotic moments. Connections with earlier results and extensions are indicated.

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