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.

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

1974 ◽  
Vol 11 (03) ◽  
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 t 2, 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.

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.

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

1972 ◽  
Vol 9 (04) ◽  
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.

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.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (04) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For everyi, a random numberNiof particles join the system at time. These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f.and life time distributionG(t). Assume. Then it is shown thatZ(t)e–αtconverges in distribution to an extended real-valued random variableYwhereais the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

A local limit theorem for the critical age-dependent branching process

1978 ◽  
Vol 15 (01) ◽  
pp. 46-53
Author(s):  
David L. Quigg
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Age Dependent ◽  
Critical Age ◽  
Number Of Particles

Let Z(t) denote the number of particles alive at time t in a critical age-dependent branching process. It is proved that, for k ≧ 1, there exists a constant Ak > 0 such that t 2 P(Z(t) = k)→Ak as t→∞.

Asymptotic probabilities in a critical age-dependent branching process

1976 ◽  
Vol 13 (03) ◽  
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→∞ t 2 P[Z(t) = k] = ak > 0, where {ak } are constants. The method is also applied to the total progeny in the critical process.

Sign in / Sign up

Export Citation Format

Share Document