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.

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.

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> 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 <a≤ Φ ≤ β < ∞, for some constantsa, β, thenw.p.l, whereZtis the number of particles at timet.

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.

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
Image Position

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

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.

Labelling experiments and phase-age distributions for multiphase branching processes

1973 ◽  
Vol 10 (4) ◽  
pp. 739-747 ◽  
Author(s):  
P. J. Brockwell ◽  
W. H. Kuo
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Joint Probability ◽  
Random Variable ◽  
Cell Labelling ◽  
Single Individual ◽  
Age Dependent ◽  
The Individual ◽  
Number Of Individuals ◽  
Joint Probability Density

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x1, x2, x3, x4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z(4) (a0,…, an, t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y01,…, Y04,…, Yi1,…, Yi4,…, Yn4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij, i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

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.

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.

Maximal branching processes and ‘long-range percolation’

1970 ◽  
Vol 7 (01) ◽  
pp. 89-98
Author(s):  
John Lamperti
Keyword(s):  
Probability Distribution Function ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Transition Matrices ◽  
A Chain ◽  
Image Position ◽  
The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

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 &gt; 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.

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