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.

Limit theorems for stochastic growth models. II

1972 ◽  
Vol 4 (03) ◽  
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| = 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 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.

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. I

1972 ◽  
Vol 4 (2) ◽  
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| = 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→∞Znρ−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 The speed of extinction for some generalized jiřina processes

2009 ◽  
Vol 41 (2) ◽  
pp. 576-599 ◽  
Author(s):  
Yuqiang Li
Keyword(s):  
Growth Models ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Stochastic Growth ◽  
Mild Conditions ◽  
Suitable Constant ◽  
Stochastic Growth Models ◽  
Drift Parameter

The speed of extinction for some generalized Jiřina processes {Xn} is discussed. We first discuss the geometric speed. Under some mild conditions, the results reveal that the sequence {cn}, where c does not equal the pseudo-drift parameter at x = 0, cannot estimate the speed of extinction accurately. Then the general case is studied. We determine a group of sufficient conditions such that Xn/cn, with a suitable constant cn, converges almost surely as n → ∞ to a proper, nondegenerate random variable. The main tools used in this paper are exponent martingales and stochastic growth models.

scholarly journals The speed of extinction for some generalized jiřina processes

2009 ◽  
Vol 41 (02) ◽  
pp. 576-599 ◽  
Author(s):  
Yuqiang Li
Keyword(s):  
Growth Models ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Stochastic Growth ◽  
Mild Conditions ◽  
Suitable Constant ◽  
Stochastic Growth Models ◽  
Drift Parameter

The speed of extinction for some generalized Jiřina processes {X n } is discussed. We first discuss the geometric speed. Under some mild conditions, the results reveal that the sequence {c n }, where c does not equal the pseudo-drift parameter at x = 0, cannot estimate the speed of extinction accurately. Then the general case is studied. We determine a group of sufficient conditions such that X n /c n , with a suitable constant c n , converges almost surely as n → ∞ to a proper, nondegenerate random variable. The main tools used in this paper are exponent martingales and stochastic growth models.

A renewal density theorem in the multi-dimensional case

1967 ◽  
Vol 4 (1) ◽  
pp. 62-76 ◽  
Author(s):  
Charles J. Mode
Keyword(s):  
Density Function ◽  
Covariance Matrix ◽  
Branching Processes ◽  
Random Variable ◽  
Density Theorem ◽  
Positive Definite ◽  
Dimensional Case ◽  
Age Dependent ◽  
Image Position ◽  
Mean Vector

SummaryIn this note a renewal density theorem in the multi-dimensional case is formulated and proved. Let f(x) be the density function of a p-dimensional random variable with positive mean vector μ and positive-definite covariance matrix Σ, let hn(x) be the n-fold convolution of f(x) with itself, and set Then for arbitrary choice of integers k1, …, kp–1 distinct or not in the set (1, 2, …, p), it is shown that under certain conditions as all elements in the vector x = (x1, …, xp) become large. In the above expression μ‵ is interpreted as a row vector and μ a column vector. An application to the theory of a class of age-dependent branching processes is also presented.

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