On higher moments of the population size in a subcritical branching process

1988 ◽  
Vol 25 (2) ◽  
pp. 413-417 ◽  
Author(s):  
Eric Willekens
Keyword(s):  
Rate Of Convergence ◽  
Population Size ◽  
Branching Process ◽  
Convergence Result ◽  
Lifetime Distribution ◽  
Higher Moments ◽  
Age Dependent

Let {Z(t), t ≧ 0} be an age-dependent subcritical branching process. In this paper we show that if the lifetime distribution is subexponential, EZα (t) ~ EZ(t) (t →∞) for every α ≧ 1. If furthermore the lifetime distribution has a subexponential density, a rate of convergence result in the above relation is established.

On higher moments of the population size in a subcritical branching process

1988 ◽  
Vol 25 (02) ◽  
pp. 413-417
Author(s):  
Eric Willekens
Keyword(s):  
Rate Of Convergence ◽  
Population Size ◽  
Branching Process ◽  
Convergence Result ◽  
Lifetime Distribution ◽  
Higher Moments ◽  
Age Dependent

Let {Z(t), t ≧ 0} be an age-dependent subcritical branching process. In this paper we show that if the lifetime distribution is subexponential, EZα (t) ~ EZ(t) (t →∞) for every α ≧ 1. If furthermore the lifetime distribution has a subexponential density, a rate of convergence result in the above relation is established.

Sums of Lifetimes in Age Dependent Branching Processes

1969 ◽  
Vol 6 (01) ◽  
pp. 195-200 ◽  
Author(s):  
J. Howard Weiner
Keyword(s):  
Distribution Function ◽  
Branching Process ◽  
Past History ◽  
Branching Processes ◽  
Lifetime Distribution ◽  
Parent Cell ◽  
Absolutely Continuous ◽  
Age Dependent ◽  
A Cell ◽  
Additional Assumptions

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σ k=0 ∞ pk s k and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

The asymptotic behaviour of extinction probability in the Smith–Wilkinson branching process

1993 ◽  
Vol 25 (02) ◽  
pp. 263-289
Author(s):  
D. R. Grey ◽  
Lu Zhunwei
Keyword(s):  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Random Environment ◽  
Critical State ◽  
Branching Process ◽  
Extinction Probability ◽  
Subcritical State ◽  
Regularity Conditions

Under some regularity conditions, in the supercritical Smith–Wilkinson branching process it is shown that as k, the starting population size, tends to infinity, the rate of convergence of qk, the corresponding extinction probability, to zero is similar to that of: k–θ, if there exists at least one subcritical state in the random environment space; xkk–α , if there exist only supercritical states in the random environment space; exp , if there exists at least one critical state and the others are supercritical in the random environment space. Here θ, x, α and c are positive constants determined by the process.

Asymptotic probabilities in a critical age-dependent branching process

1976 ◽  
Vol 13 (3) ◽  
pp. 476-485 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Branching Process ◽  
Comparison Method ◽  
Lifetime Distribution ◽  
Order Moment ◽  
Moment Condition ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The Mean ◽  
Critical Process

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.The method is also applied to the total progeny in the critical process.

scholarly journals Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process

1969 ◽  
Vol 10 (1-2) ◽  
pp. 231-235 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Growth Rate ◽  
Population Size ◽  
Branching Process ◽  
Expected Number ◽  
Lattice Distribution ◽  
Asymptotic Growth ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
Asymptotic Growth Rate

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

On a functional equation for general branching processes

1973 ◽  
Vol 10 (1) ◽  
pp. 198-205 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Population Size ◽  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Slow Variation ◽  
Age Dependent ◽  
Class Of Functions

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.

Sums of Lifetimes in Age Dependent Branching Processes

1969 ◽  
Vol 6 (1) ◽  
pp. 195-200 ◽  
Author(s):  
J. Howard Weiner
Keyword(s):  
Distribution Function ◽  
Branching Process ◽  
Past History ◽  
Branching Processes ◽  
Lifetime Distribution ◽  
Parent Cell ◽  
Absolutely Continuous ◽  
Age Dependent ◽  
A Cell ◽  
Additional Assumptions

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σk=0∞pksk and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

The asymptotic behaviour of extinction probability in the Smith–Wilkinson branching process

1993 ◽  
Vol 25 (2) ◽  
pp. 263-289 ◽  
Author(s):  
D. R. Grey ◽  
Lu Zhunwei
Keyword(s):  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Random Environment ◽  
Critical State ◽  
Branching Process ◽  
Extinction Probability ◽  
Subcritical State ◽  
Regularity Conditions

Under some regularity conditions, in the supercritical Smith–Wilkinson branching process it is shown that as k, the starting population size, tends to infinity, the rate of convergence of qk, the corresponding extinction probability, to zero is similar to that of:k–θ, if there exists at least one subcritical state in the random environment space; xkk–α, if there exist only supercritical states in the random environment space; exp , if there exists at least one critical state and the others are supercritical in the random environment space.Here θ, x, α and c are positive constants determined by the process.

On the asymptotic properties of a supercritical bisexual branching process

1986 ◽  
Vol 23 (03) ◽  
pp. 820-826 ◽  
Author(s):  
J. H. Bagley
Keyword(s):  
Population Size ◽  
General Class ◽  
Population Model ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Convergence Result ◽  
Bisexual Population

An almost sure convergence result for the normed population size of a bisexual population model is proved. Properties of the limit random variable are deduced. The derivation of similar results for a general class of such processes is discussed.

An age dependent branching process with variable lifetime distribution: The generation size

1974 ◽  
Vol 6 (02) ◽  
pp. 291-308 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
The Mean ◽  
Watson Process ◽  
Number Of Particles

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk (t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk (t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

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