Space-time harmonic functions and age-dependent branching processes

1975 ◽  
Vol 7 (2) ◽  
pp. 283-298 ◽  
Author(s):  
Thomas H. Savits
Keyword(s):  
Harmonic Functions ◽  
Positive Root ◽  
Branching Processes ◽  
Random Variable ◽  
Space Time ◽  
Malthusian Parameter ◽  
Time Harmonic ◽  
Age Dependent ◽  
Image Position ◽  
Number Of Particles

Let X be an age-dependent branching process with lifetime distribution G and age-dependent generating function π(y,s) = σk = 0∞pk(y) sk. We assume that G is right-continuous and G(0+) = G(0) = 0. The base state space S is [0,T) where T = inf{t : G(t) = 1}. Set m(y) = σk = 0∞k pk(y) and Then extinction occurs with probability one iff m ≤ 1. In the case where m > 1, define the Malthusian parameter λ to be the unique (positive) root of and set on S. is a -space-time harmonic function of the process X and the corresponding non-negative martingale converges w.p.l to a random variable W; furthermore, under a regularity assumption, W is non-trivial iff where and If 0 < a ≤ Φ ≤ β < ∞, for some constants a, β, then w.p.l, where Zt is the number of particles at time t.

Space-time harmonic functions and age-dependent branching processes

1975 ◽  
Vol 7 (02) ◽  
pp. 283-298
Author(s):  
Thomas H. Savits
Keyword(s):  
Harmonic Functions ◽  
Positive Root ◽  
Branching Processes ◽  
Random Variable ◽  
Space Time ◽  
Malthusian Parameter ◽  
Time Harmonic ◽  
Age Dependent ◽  
Image Position ◽  
Number Of Particles

LetXbe an age-dependent branching process with lifetime distributionGand age-dependent generating function π(y,s) = σk= 0∞pk(y)sk. We assume thatGis right-continuous andG(0+) =G(0) = 0. The base state spaceSis [0,T) whereT= inf{t:G(t) = 1}. Setm(y) = σk= 0∞k pk(y) andThen extinction occurs with probability one iffm≤ 1. In the case wherem&gt; 1, define the Malthusian parameter λ to be the unique (positive) root ofand setonS.is a-space-time harmonic function of the processXand the corresponding non-negative martingaleconverges w.p.l to a random variableW; furthermore, under a regularity assumption,Wis non-trivial iffwhereandIf 0 &lt;a≤ Φ ≤ β &lt; ∞, for some constantsa, β, thenw.p.l, whereZtis the number of particles at timet.

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.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

A renewal density theorem in the multi-dimensional case

1967 ◽  
Vol 4 (01) ◽  
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

Summary In 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 k 1, …, 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 = (x 1, …, 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.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (04) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For everyi, a random numberNiof particles join the system at time. These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f.and life time distributionG(t). Assume. Then it is shown thatZ(t)e–αtconverges in distribution to an extended real-valued random variableYwhereais the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

scholarly journals Attainable bounds for expectations

1982 ◽  
Vol 33 (3) ◽  
pp. 411-420 ◽  
Author(s):  
E. Seneta ◽  
N. C. Weber
Keyword(s):  
Lower Bound ◽  
Generating Functions ◽  
Branching Process ◽  
Simple Technique ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Coefficient Of Skewness

AbstractA simple technique for obtaining bounds in terms of means and variances for the expectations of certain functions of random variables in a given class is examined. The bounds given are sharp in the sense that they are attainable by at least one random variable in the class. This technique is applied to obtain bounds for moment generating functions, the coefficient of skewness and parameters associated with branching processes. In particular an improved lower bound for the Malthusian parameter in an age-dependent branching process is derived.

Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

1976 ◽  
Vol 13 (3) ◽  
pp. 455-465
Author(s):  
D. I. Saunders
Keyword(s):  
State Space ◽  
Branching Process ◽  
Branching Processes ◽  
Rates Of Convergence ◽  
Expected Number ◽  
Subcritical Case ◽  
Arbitrary State ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

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