scholarly journals On the Maximal Offspring in a Critical Branching Process with Infinite Variance

2011 ◽  
Vol 48 (2) ◽  
pp. 576-582 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Branching Process ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Infinite Variance ◽  
Regularly Varying ◽  
Watson Process ◽  
Critical Branching Process

We investigate the maximal number Mk of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k-1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

scholarly journals On the Maximal Offspring in a Critical Branching Process with Infinite Variance

2011 ◽  
Vol 48 (02) ◽  
pp. 576-582 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Branching Process ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Infinite Variance ◽  
Regularly Varying ◽  
Watson Process ◽  
Critical Branching Process

We investigate the maximal number M k of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 &lt; α &lt; 2, then k -1 M k converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

On Largest Offspring in a Critical Branching Process with Finite Variance

2013 ◽  
Vol 50 (03) ◽  
pp. 791-800 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Shape Parameter ◽  
Weak Limit ◽  
Finite Variance ◽  
Poisson Measure ◽  
Doubly Stochastic ◽  
Weak Limit Theorem ◽  
Watson Process ◽  
Critical Branching Process

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X * k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α &gt; 2 (and, hence, finite variance). We show that X * k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1&lt; α&lt;2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

scholarly journals On Largest Offspring in a Critical Branching Process with Finite Variance

2013 ◽  
Vol 50 (3) ◽  
pp. 791-800 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Shape Parameter ◽  
Weak Limit ◽  
Finite Variance ◽  
Poisson Measure ◽  
Doubly Stochastic ◽  
Weak Limit Theorem ◽  
Watson Process ◽  
Critical Branching Process

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X*k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α > 2 (and, hence, finite variance). We show that X*k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

scholarly journals On estimation of the variances for critical branching processes with immigration

2010 ◽  
Vol 47 (02) ◽  
pp. 526-542
Author(s):  
Chunhua Ma ◽  
Longmin Wang
Keyword(s):  
Least Squares ◽  
Branching Process ◽  
Branching Processes ◽  
Limiting Distributions ◽  
Least Squares Estimators ◽  
Conditional Least Squares ◽  
Regularly Varying ◽  
Branching Process With Immigration ◽  
Critical Branching Process

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 &lt; α &lt; 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

Remarks on the maxima of a martingale sequence with application to the simple critical branching process

1987 ◽  
Vol 24 (03) ◽  
pp. 768-772 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Branching Process ◽  
Alternative Proof ◽  
Watson Process ◽  
Image Position ◽  
Critical Branching Process

Let where {Zn, ℱn } is a non-negative submartingale satisfying Ei (Zn log Zn ) →∞. It is shown that When {Zn } is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

The critical branching process with immigration stopped at zero

1985 ◽  
Vol 22 (01) ◽  
pp. 223-227 ◽  
Author(s):  
B. Gail Ivanoff ◽  
E. Seneta
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Critical Case ◽  
Initial Distribution ◽  
New Form ◽  
Watson Process ◽  
Branching Process With Immigration ◽  
Limit Result ◽  
Critical Branching Process

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).

Remarks on the maxima of a martingale sequence with application to the simple critical branching process

1987 ◽  
Vol 24 (3) ◽  
pp. 768-772 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Branching Process ◽  
Alternative Proof ◽  
Watson Process ◽  
Image Position ◽  
Critical Branching Process

Let where {Zn, ℱn} is a non-negative submartingale satisfying Ei(Zn log Zn) →∞. It is shown that When {Zn} is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

The critical branching process with immigration stopped at zero

1985 ◽  
Vol 22 (1) ◽  
pp. 223-227 ◽  
Author(s):  
B. Gail Ivanoff ◽  
E. Seneta
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Critical Case ◽  
Initial Distribution ◽  
New Form ◽  
Watson Process ◽  
Branching Process With Immigration ◽  
Limit Result ◽  
Critical Branching Process

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).

A Galton–Watson process with a threshold

2016 ◽  
Vol 53 (2) ◽  
pp. 614-621
Author(s):  
K. B. Athreya ◽  
H.-J. Schuh
Keyword(s):  
Positive Integer ◽  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Mean Value ◽  
Size Dependent ◽  
The Mean ◽  
Watson Process ◽  
Critical Branching Process

Abstract In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

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