A conditional limit theorem for a critical Branching process with immigration

1977 ◽  
Vol 21 (5) ◽  
pp. 405-411 ◽  
Author(s):  
V. A. Vatutin
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Conditional Limit Theorem ◽  
Branching Process With Immigration ◽  
Conditional Limit ◽  
Critical Branching Process

scholarly journals The Critical Galton-Watson Process Without Further Power Moments

2007 ◽  
Vol 44 (03) ◽  
pp. 753-769 ◽  
Author(s):  
S. V. Nagaev ◽  
V. Wachtel
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Alternative Proof ◽  
Conditional Limit Theorem ◽  
Power Moments ◽  
Watson Process ◽  
Conditional Limit

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

scholarly journals Berry-Esseen bound for nearly critical branching processes with immigration

2019 ◽  
pp. 42-49
Author(s):  
Ya. Khusanbaev ◽  
S. Sharipov ◽  
V. Golomoziy
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Branching Process ◽  
Branching Processes ◽  
Branching Process With Immigration ◽  
Critical Branching Process

In this paper, we consider a nearly critical branching process with immigration. We obtain the rate of convergence in central limit theorem for nearly critical branching processes with immigration.

scholarly journals The Critical Galton-Watson Process Without Further Power Moments

2007 ◽  
Vol 44 (3) ◽  
pp. 753-769 ◽  
Author(s):  
S. V. Nagaev ◽  
V. Wachtel
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Alternative Proof ◽  
Conditional Limit Theorem ◽  
Power Moments ◽  
Watson Process ◽  
Conditional Limit

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Zn; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

Efficiency, Sufficiency, and Optimality

2017 ◽  
Author(s):  
Amos Golan
Keyword(s):  
Limit Theorem ◽  
Computational Efficiency ◽  
The Other ◽  
Information Compression ◽  
Information Theoretic ◽  
Conditional Limit Theorem ◽  
Concept Of Information ◽  
Conditional Limit

In this chapter I provide additional rationalization for using the info-metrics framework. This time the justifications are in terms of the statistical, mathematical, and information-theoretic properties of the formalism. Specifically, in this chapter I discuss optimality, statistical and computational efficiency, sufficiency, the concentration theorem, the conditional limit theorem, and the concept of information compression. These properties, together with the other properties and measures developed in earlier chapters, provide logical, mathematical, and statistical justifications for employing the info-metrics framework.

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.

Multitype processes with reproduction-dependent immigration

1998 ◽  
Vol 35 (02) ◽  
pp. 281-292
Author(s):  
Ibrahim Rahimov
Keyword(s):  
Limit Theorem ◽  
Discrete Time ◽  
Branching Process ◽  
Martingale Approach ◽  
Offspring Distribution ◽  
Second Moments ◽  
Branching Process With Immigration

The multitype discrete time indecomposable branching process with immigration is considered. Using a martingale approach a limit theorem is proved for such processes when the totality of immigrating individuals at a given time depends on evolution of the processes generating by previously immigrated individuals. Corollaries of the limit theorem are obtained for the cases of finite and infinite second moments of offspring distribution in critical processes.

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).

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