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.

scholarly journals Conditional limit theorems for branching processes

1991 ◽  
Vol 4 (4) ◽  
pp. 263-292 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Conditional Limit Theorems ◽  
Total Progeny ◽  
Number Of Individuals ◽  
Conditional Limit

Let [ξ(m),m=0,1,2,…] be a branching process in which each individual reproduces independently of the others and has probability pj(j=0,1,2,…) of giving rise to j descendants in the following generation. The random variable ξ(m) is the number of individuals in the mth generation. It is assumed that P{ξ(0)=1}=1. Denote by ρ the total progeny, μ, the time of extinction, and τ, the total number of ancestors of all the individuals in the process. This paper deals with the distributions of the random variables ξ(m), μ and τ under the condition that ρ=n and determines the asymptotic behavior of these distributions in the case where n→∞ and m→∞ in such a way that m/n tends to a finite positive limit.

The total progeny of a simple branching process with state-dependent immigration

1983 ◽  
Vol 20 (03) ◽  
pp. 472-481 ◽  
Author(s):  
M. V. Kulkarni ◽  
A. G. Pakes
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
State Dependent ◽  
Total Progeny

The paper establishes limit theorems for the total progeny of a simple branching process in which immigration is allowed whenever the population size reaches 0.

The total progeny of a simple branching process with state-dependent immigration

1983 ◽  
Vol 20 (3) ◽  
pp. 472-481 ◽  
Author(s):  
M. V. Kulkarni ◽  
A. G. Pakes
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
State Dependent ◽  
Total Progeny

The paper establishes limit theorems for the total progeny of a simple branching process in which immigration is allowed whenever the population size reaches 0.

Some limit theorems for the total progeny of a branching process

1971 ◽  
Vol 3 (01) ◽  
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 a j (j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 &lt; a 0, a 0 + a 1 &lt; 1.

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.

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.

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&gt; 0 and for suitableq(·), these theorems can be translated into limit theorems forX(t), by using the continuous mapping theorem.

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