Critical branching processes

1974 ◽  
Vol 6 (03) ◽  
pp. 434-445 ◽  
Author(s):  
Peter Ney
Keyword(s):  
Branching Processes ◽  
Comparison Method ◽  
Moment Conditions ◽  
Second Moment ◽  
Dependent Process ◽  
Age Dependent

This paper develops a comparison method for critical branching processes. The method is applied to prove the exponential limit law for the multi-type age-dependent process under second moment conditions.

Critical branching processes

1974 ◽  
Vol 6 (3) ◽  
pp. 434-445 ◽  
Author(s):  
Peter Ney
Keyword(s):  
Branching Processes ◽  
Comparison Method ◽  
Moment Conditions ◽  
Second Moment ◽  
Dependent Process ◽  
Age Dependent

This paper develops a comparison method for critical branching processes. The method is applied to prove the exponential limit law for the multi-type age-dependent process under second moment conditions.

A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v.It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1]k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant.This result is then used to give a new proof of the exponential limit law.

A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (02) ◽  
pp. 235-242
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z 1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v. It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1] k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant. This result is then used to give a new proof of the exponential limit law.

Multi-type age-dependent branching processes as models of metastasis evolution

2019 ◽  
Vol 35 (3) ◽  
pp. 284-299
Author(s):  
Maroussia Slavtchova-Bojkova ◽  
Kaloyan Vitanov
Keyword(s):  
Branching Processes ◽  
Age Dependent

PULP STONES: A REVIEW

2021 ◽  
pp. 48-50
Author(s):  
Eva Shukla ◽  
Shruthi Nagaraja
Keyword(s):  
Historical Perspective ◽  
Review Article ◽  
Young Population ◽  
Etiological Factors ◽  
Dependent Process ◽  
Age Dependent ◽  
Pulp Stones

Pulp stones are seen to be a fairly common nding in most routine radiographs but they have been given relatively less attention in research as well as textbooks. New investigations have been done in abundance when it comes to their prevalence as well as etiological factors. Despite numerous studies, their formation still remains a riddle for endodontists and researchers alike. Historical perspective of pulp stones being an age dependent process are now being questioned as more studies report their prevalence in young population. This review article evaluates the recent studies pertaining to pulpal calcications where newer etiological factors as well as molecular factors have been highlighted in their formation. Their signicance has been largely studied correlating their occurrence with various other systemic calcications.

Stochastic models in cell kinetics

1988 ◽  
Vol 25 (A) ◽  
pp. 91-111
Author(s):  
Peter J. Brockwell
Keyword(s):  
Life Cycle ◽  
Stochastic Models ◽  
Death Rate ◽  
Cell Kinetics ◽  
Branching Processes ◽  
Experimental Studies ◽  
Renewal Theory ◽  
Biological Cell ◽  
Age Dependent

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.

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.

Two-dimensional renewal theorems with weak moment conditions and critical Bellman – Harris branching processes

2016 ◽  
Vol 26 (1) ◽  
Author(s):  
Valentin A. Topchiy
Keyword(s):  
Branching Processes ◽  
Lifetime Distribution ◽  
Two Dimensional ◽  
Moment Conditions

AbstractWe consider critical Bellman – Harris processes with two types of particles. The tail of the lifetime distribution of the first type particles decreases as

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