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.

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.

Stochastic growth models: bounds on critical values

1992 ◽  
Vol 29 (1) ◽  
pp. 11-20 ◽  
Author(s):  
R. Durrett
Keyword(s):  
Interacting Particle Systems ◽  
Growth Models ◽  
Upper Bounds ◽  
Particle Systems ◽  
Critical Values ◽  
Oriented Percolation ◽  
Finite Sets ◽  
Stochastic Growth ◽  
Interacting Particle ◽  
Stochastic Growth Models

We give upper bounds on the critical values for oriented percolation and some interacting particle systems by computing their behavior on small finite sets.

scholarly journals Fingering in Stochastic Growth Models

2014 ◽  
Vol 23 (4) ◽  
pp. 465-474 ◽  
Author(s):  
Andreas C. Aristotelous ◽  
Richard Durrett
Keyword(s):  
Growth Models ◽  
Stochastic Growth ◽  
Stochastic Growth Models

scholarly journals Mathematical and Stochastic Growth Models

2021 ◽  
Vol 9 (11) ◽  
pp. 1576-1582
Author(s):  
V. Pavankumari
Keyword(s):  
Mathematical Models ◽  
Exponential Growth ◽  
Growth Models ◽  
Future Research ◽  
Applied Statistics ◽  
Stochastic Growth ◽  
Research Problems ◽  
Stochastic Growth Models

Abstract: Many statistical and mathematical models of growth are developed in the literature and effectively applied to various conditions in the existent world that involve many research problems in the different fields of applied statistics. Nevertheless, still, there is an equally large number of conditions, which have not yet been mathematically or statistically modeled, due to the complex situations or formed models are mathematically or statistically inflexible. The present study is based on mathematical and stochastic growth models. The specification of both the growth models is depicted. A detailed study of newly modified growth models is mentioned. This research will give substantial information on growth models, such as proposed modified exponential growth models and their specifications clearly motioned which gives scope for future research.

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