Asymptotic properties of supercritical branching processes I: The Galton-Watson process

1974 ◽  
Vol 6 (4) ◽  
pp. 711-731 ◽  
Author(s):  
N. H. Bingham ◽  
R. A. Doney
Keyword(s):  
Regular Variation ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
The Other ◽  
Integer Exponent ◽  
Watson Process ◽  
Supercritical Branching Processes

We obtain results connecting the distributions of the random variables Z1 and W in the supercritical Galton-Watson process. For example, if a > 1, and converge or diverge together, and regular variation of the tail of one of Z1, W with non-integer exponent α > 1 is equivalent to regular variation of the tail of the other.

Asymptotic properties of supercritical branching processes I: The Galton-Watson process

1974 ◽  
Vol 6 (04) ◽  
pp. 711-731 ◽  
Author(s):  
N. H. Bingham ◽  
R. A. Doney
Keyword(s):  
Regular Variation ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
The Other ◽  
Integer Exponent ◽  
Watson Process ◽  
Supercritical Branching Processes

We obtain results connecting the distributions of the random variables Z 1 and W in the supercritical Galton-Watson process. For example, if a > 1, and converge or diverge together, and regular variation of the tail of one of Z 1, W with non-integer exponent α > 1 is equivalent to regular variation of the tail of the other.

Asymptotic properties of super-critical branching processes II: Crump-Mode and Jirina processes

1975 ◽  
Vol 7 (01) ◽  
pp. 66-82 ◽  
Author(s):  
N. H. Bingham ◽  
R. A. Doney
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Regular Variation ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
The Other ◽  
Continuous State Space ◽  
Continuous State ◽  
Integer Exponent

We obtain results connecting the distribution of the random variablesYandWin the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1,EYβandEWβconverge or diverge together and regular variation of the tail of one ofY, Wwith non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

Asymptotic properties of super-critical branching processes II: Crump-Mode and Jirina processes

1975 ◽  
Vol 7 (1) ◽  
pp. 66-82 ◽  
Author(s):  
N. H. Bingham ◽  
R. A. Doney
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Regular Variation ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
The Other ◽  
Continuous State Space ◽  
Continuous State ◽  
Integer Exponent

We obtain results connecting the distribution of the random variables Y and W in the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1, EYβ and EWβ converge or diverge together and regular variation of the tail of one of Y, W with non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

Limit theorems and estimation theory for branching processes with an increasing random number of ancestors

1997 ◽  
Vol 34 (2) ◽  
pp. 309-327 ◽  
Author(s):  
J. P. Dion ◽  
N. M. Yanev
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Number ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Transfer Theorem ◽  
Asymptotically Normal ◽  
The Mean ◽  
Watson Process

This paper deals with a Bienaymé-Galton-Watson process having a random number of ancestors. Its asymptotic properties are studied when both the number of ancestors and the number of generations tend to infinity. This yields consistent and asymptotically normal estimators of the mean and the offspring distribution of the process. By exhibiting a connection with the BGW process with immigration, all results can be transported to the immigration case, under an appropriate sampling scheme. A key feature of independent interest is a new limit theorem for sums of a random number of random variables, which extends the Gnedenko and Fahim (1969) transfer theorem.

Limit theorems and estimation theory for branching processes with an increasing random number of ancestors

1997 ◽  
Vol 34 (02) ◽  
pp. 309-327 ◽  
Author(s):  
J. P. Dion ◽  
N. M. Yanev
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Number ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Transfer Theorem ◽  
Asymptotically Normal ◽  
The Mean ◽  
Watson Process

This paper deals with a Bienaymé-Galton-Watson process having a random number of ancestors. Its asymptotic properties are studied when both the number of ancestors and the number of generations tend to infinity. This yields consistent and asymptotically normal estimators of the mean and the offspring distribution of the process. By exhibiting a connection with the BGW process with immigration, all results can be transported to the immigration case, under an appropriate sampling scheme. A key feature of independent interest is a new limit theorem for sums of a random number of random variables, which extends the Gnedenko and Fahim (1969) transfer theorem.

Integer-valued branching processes with immigration

1983 ◽  
Vol 15 (04) ◽  
pp. 713-725 ◽  
Author(s):  
F. W. Steutel ◽  
W. Vervaat ◽  
S. J. Wolfe
Keyword(s):  
Difference Equation ◽  
Continuous Time ◽  
Queueing Theory ◽  
Branching Processes ◽  
Random Variables ◽  
Discrete State ◽  
Limiting Behaviour ◽  
Supercritical Branching Processes

The notion of self-decomposability for -valued random variables as introduced by Steutel and van Harn [10] and its generalization by van Harn, Steutel and Vervaat [5], are used to study the limiting behaviour of continuous-time Markov branching processes with immigration. This behaviour provides analogues to the behaviour of sequences of random variables obeying a certain difference equation as studied by Vervaat [12] and their continuous-time counterpart considered by Wolfe [13]. An application in queueing theory is indicated. Furthermore, discrete-state analogues are given for results on stability in the processes studied by Wolfe, and for results on self-decomposability in supercritical branching processes by Yamazato [14].

On a theorem of bingham and doney

1982 ◽  
Vol 19 (01) ◽  
pp. 217-220 ◽  
Author(s):  
A. De Meyer
Keyword(s):  
Regular Variation ◽  
Random Variables ◽  
Watson Process

We obtain an extension of a theorem of Bingham and Doney connecting the random variables Z 1 and W in the supercritical Galton-Watson process. The regular variation of the distribution of Z1 is equivalent to the regular variation of the tail of the distribution of W for integer values of α > 1.

A limit theorem for a class of supercritical branching processes

1972 ◽  
Vol 9 (04) ◽  
pp. 707-724 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Generating Functional ◽  
Number Of Children ◽  
Supercritical Case ◽  
Age Dependent ◽  
Watson Process ◽  
Supercritical Branching Processes ◽  
Convergence In Mean

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn } form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ 1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn /mn to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].

Collapsing supercritical branching processes

1979 ◽  
Vol 16 (04) ◽  
pp. 732-739 ◽  
Author(s):  
Fergus Daly
Keyword(s):  
Branching Processes ◽  
Population Processes ◽  
Watson Process ◽  
Supercritical Branching Processes

Characteristics of some collapsing supercritical stochastic population processes where an embedded Galton–Watson process can be isolated are shown to be identical to those of some associated subcritical process.

On maximum family size in branching processes

1999 ◽  
Vol 36 (3) ◽  
pp. 632-643 ◽  
Author(s):  
Ibrahim Rahimov ◽  
George P. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Sample Size ◽  
Random Sample ◽  
Family Size ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Sample Size ◽  
Convergence Results ◽  
Watson Process ◽  
Supercritical Branching Processes

The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

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