A critical branching process with immigration in varying environments

2021 ◽  
Vol 168 ◽  
pp. 108928
Author(s):  
Kosto V. Mitov
Keyword(s):  
Branching Process ◽  
Varying Environments ◽  
Branching Process With Immigration ◽  
Critical Branching Process

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 < α < 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

Asymptotic Expansions for Array Branching Processes with Applications to Bootstrapping

1998 ◽  
Vol 35 (1) ◽  
pp. 12-26 ◽  
Author(s):  
T. N. Sriram
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Asymptotic Expansions ◽  
Branching Process ◽  
Branching Processes ◽  
Test Function ◽  
Branching Process With Immigration ◽  
Critical Branching Process

Asymptotic expansions are obtained for the distribution function of a studentized estimator of the offspring mean sequence in an array branching process with immigration. The expansion result is shown to hold in a test function topology. As an application of this result, it is shown that the bootstrapping distribution of the estimator of the offspring mean in a sub-critical branching process with immigration also admits the same expansion (in probability). From these considerations, it is concluded that the bootstrapping distribution provides a better approximation asymptotically than the normal distribution.

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).

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).

scholarly journals Branching processes in a random environment with immigration stopped at zero

2020 ◽  
Vol 57 (1) ◽  
pp. 237-249 ◽  
Author(s):  
Elena Dyakonova ◽  
Doudou Li ◽  
Vladimir Vatutin ◽  
Mei Zhang
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Time Interval ◽  
Tail Distribution ◽  
The Moment ◽  
Branching Process With Immigration ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

AbstractA critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.

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