Limit theorems for critical randomly indexed branching processes

2010 ◽  
pp. 95-108 ◽  
Author(s):  
Kosto V. Mitov ◽  
Georgi K. Mitov ◽  
Nikolay M. Yanev
Keyword(s):  
Limit Theorems ◽  
Branching Processes

scholarly journals Limit theorems for multi-type general branching processes with population dependence

2020 ◽  
Vol 52 (4) ◽  
pp. 1127-1163
Author(s):  
Jie Yen Fan ◽  
Kais Hamza ◽  
Peter Jagers ◽  
Fima C. Klebaner
Keyword(s):  
Carrying Capacity ◽  
Population Model ◽  
General Framework ◽  
Limit Theorems ◽  
Mating Systems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Special Section ◽  
Large Numbers ◽  
Type Population

AbstractA general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.

scholarly journals Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration

2017 ◽  
Vol 54 (2) ◽  
pp. 569-587 ◽  
Author(s):  
Ollivier Hyrien ◽  
Kosto V. Mitov ◽  
Nikolay M. Yanev
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Cell Populations ◽  
Subcritical Case ◽  
Immigration Process ◽  
Large Numbers

Abstract We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.

RANDOM WALKS WITH HEAVY TAILS AND LIMIT THEOREMS FOR BRANCHING PROCESSES WITH MIGRATION AND IMMIGRATION

2012 ◽  
Vol 12 (01) ◽  
pp. 1150007 ◽  
Author(s):  
YAQIN FENG ◽  
STANISLAV MOLCHANOV ◽  
JOSEPH WHITMEYER
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Heavy Tails ◽  
Stochastic Dynamics ◽  
Branching Processes ◽  
Spatial Dynamics ◽  
Limiting Distributions ◽  
Central Result

The central result of this paper is the existence of limiting distributions for two classes of critical homogeneous-in-space branching processes with heavy tails spatial dynamics in dimension d = 2. In dimension d ≥ 3, the same results are true without any special assumptions on the underlying (non-degenerated) stochastic dynamics.

Limit theorems for stochastic growth models. I

1972 ◽  
Vol 4 (02) ◽  
pp. 193-232 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Fixed Point ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Growth Models ◽  
Branching Processes ◽  
Random Variable ◽  
Present Situation ◽  
Stochastic Growth ◽  
Stochastic Growth Models ◽  
Zygotic Selection

We consider d-dimensional stochastic processes which take values in (R+) d . These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some Here τ: (R+) d → R+, |x| = Σ1 d |x(i)| A = {x ∈ (R+) d : |x| = 1} and T: A → A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that lim n→∞ Z n ρ−n = wp. This result is a generalization of the main limit theorem for super-critical branching processes; note, however, that in the present situation both p and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

scholarly journals Limit Theorems for the Split Times of Branching Processes

1967 ◽  
Vol 17 (3) ◽  
pp. 257-277
Author(s):  
Krishna Athreya ◽  
Samuel Karlin
Keyword(s):  
Limit Theorems ◽  
Branching Processes

Limit theorems for stochastic growth models. II

1972 ◽  
Vol 4 (3) ◽  
pp. 393-428 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Limit Theorems ◽  
Growth Models ◽  
Branching Processes ◽  
Random Variable ◽  
Present Situation ◽  
Stochastic Growth ◽  
Stochastic Growth Models ◽  
Zygotic Selection ◽  
Image Position ◽  
Supercritical Branching Processes

We consider d-dimensional stochastic processes which take values in (R+)d These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some Here τ:(R+)d→R +, |x| = σ1d |x(i)|, A {x ∈(R+)d: |x| 1} and T: A→A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Zn|ρnwp. This result is a generalization of the main limit theorem for supercritical branching processes; note, however, that in the present situation both ρ and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

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