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

1974 ◽  
Vol 6 (2) ◽  
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).

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).

An age dependent branching process with variable lifetime distribution

1972 ◽  
Vol 4 (3) ◽  
pp. 453-474 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Integral Equations ◽  
Branching Process ◽  
Probability Generating Function ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Distribution Functions ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
Number Of Particles

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {Gi(t)}, together with a probability generating function, h(s) = Σk∞= 0pksk. An ith generation particle lives a random length of time, determined by Gi(t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution Gi + 1 (t).Let Zi(t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Zi(t) and it is shown that for some constants Ni, γ, a, Zi (t)/(Nitγ-1eαt) converges in mean square to a proper random variable.

An age dependent branching process with variable lifetime distribution

1972 ◽  
Vol 4 (03) ◽  
pp. 453-474 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Integral Equations ◽  
Branching Process ◽  
Probability Generating Function ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Distribution Functions ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
Number Of Particles

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {G i (t)}, together with a probability generating function, h(s) = Σ k ∞= 0pks k . An ith generation particle lives a random length of time, determined by G i (t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution G i + 1 (t). Let Z i (t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Z i (t) and it is shown that for some constants N i , γ, a, Z i (t)/(N i t γ-1 e αt ) converges in mean square to a proper random variable.

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.

The convergence of a position-dependent branching process used as an approximation to a model describing the spread of an epidemic

1976 ◽  
Vol 13 (2) ◽  
pp. 338-344 ◽  
Author(s):  
J. Radcliffe
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Markov Branching Process ◽  
Geographical Spread ◽  
Mean Square Convergence ◽  
Velocity Of Propagation ◽  
The Mean ◽  
Number Of Individuals

A supercritical position-dependent Markov branching process has been used as an approximation to a model describing the initial geographical spread of a measles epidemic (Bartlett (1956)). Let α be its Malthusian parameter, ß its velocity of propagation, Z(A, t) the number of individuals in the set A at time t, and A√(ßt) = [√(ßt) r: r ∈ A]. The mean square convergence of the random variable W(A, t)= e–αtZ(A√(ßt), t) to a limit variable W(A) is established.

Mean-square and almost-sure convergence of supercritical age-dependent branching processes

1974 ◽  
Vol 11 (04) ◽  
pp. 678-686
Author(s):  
Edgar Z. Ganuza ◽  
S. D. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Age Distribution ◽  
Almost Sure Convergence ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Mean Square Convergence ◽  
Watson Process ◽  
Successive Generations

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

Mean-square and almost-sure convergence of supercritical age-dependent branching processes

1974 ◽  
Vol 11 (4) ◽  
pp. 678-686 ◽  
Author(s):  
Edgar Z. Ganuza ◽  
S. D. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Age Distribution ◽  
Almost Sure Convergence ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Mean Square Convergence ◽  
Watson Process ◽  
Successive Generations

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

Asymptotic probabilities in a critical age-dependent branching process

1976 ◽  
Vol 13 (03) ◽  
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→∞ t 2 P[Z(t) = k] = ak > 0, where {ak } are constants. The method is also applied to the total progeny in the critical process.

The asymptotic frequencies of the types in a multitype age-dependent branching process allowing immigration

1973 ◽  
Vol 10 (03) ◽  
pp. 652-658
Author(s):  
J. Radcliffe
Keyword(s):  
Renewal Process ◽  
Branching Process ◽  
Almost Sure Convergence ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
The Mean

The mean square and almost sure convergence of W(t) = e–αt Z(t) is proved for a super-critical multitype age-dependent branching process allowing immigration at the epochs of a renewal process. It is shown that the Malthusian parameter, asymptotic frequencies of types and stationary age distributions are the same for the processes with and without immigration.

The convergence of a position-dependent branching process used as an approximation to a model describing the spread of an epidemic

1976 ◽  
Vol 13 (02) ◽  
pp. 338-344 ◽  
Author(s):  
J. Radcliffe
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Markov Branching Process ◽  
Geographical Spread ◽  
Mean Square Convergence ◽  
Velocity Of Propagation ◽  
The Mean ◽  
Number Of Individuals

A supercritical position-dependent Markov branching process has been used as an approximation to a model describing the initial geographical spread of a measles epidemic (Bartlett (1956)). Letαbe its Malthusian parameter,ßits velocity of propagation,Z(A,t) the number of individuals in the setAat timet,andA√(ßt)= [√(ßt)r:r ∈ A]. The mean square convergence of the random variableW(A, t)= e–αtZ(A√(ßt),t) to a limit variableW(A) is established.

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