scholarly journals Exponential Growth of Bifurcating Processes with Ancestral Dependence

2015 ◽  
Vol 47 (02) ◽  
pp. 545-564 ◽  
Author(s):  
Sana Louhichi ◽  
Bernard Ycart
Keyword(s):  
Growth Rate ◽  
Laplace Transform ◽  
Exponential Growth ◽  
Branching Process ◽  
Cell Kinetics ◽  
Growth Models ◽  
Branching Processes ◽  
Independent Random Variables ◽  
The Laplace Transform ◽  
Ergodicity Coefficients

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

scholarly journals Exponential Growth of Bifurcating Processes with Ancestral Dependence

2015 ◽  
Vol 47 (2) ◽  
pp. 545-564 ◽  
Author(s):  
Sana Louhichi ◽  
Bernard Ycart
Keyword(s):  
Growth Rate ◽  
Laplace Transform ◽  
Exponential Growth ◽  
Branching Process ◽  
Cell Kinetics ◽  
Growth Models ◽  
Branching Processes ◽  
Independent Random Variables ◽  
The Laplace Transform ◽  
Ergodicity Coefficients

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

scholarly journals A note on Markov branching processes

1985 ◽  
Vol 17 (02) ◽  
pp. 463-464
Author(s):  
Fred M. Hoppe
Keyword(s):  
Laplace Transform ◽  
Simple Proof ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
The Laplace Transform

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

scholarly journals A note on Markov branching processes

1985 ◽  
Vol 17 (2) ◽  
pp. 463-464 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Laplace Transform ◽  
Simple Proof ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
The Laplace Transform

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

Exponential growth of branching processes in a general context of lifetimes and birthtimes dependence

2019 ◽  
Vol 23 ◽  
pp. 584-606 ◽  
Author(s):  
Loïc Hervé ◽  
Sana Louhichi ◽  
Françoise Pène
Keyword(s):  
Laplace Transform ◽  
Exponential Growth ◽  
Spectral Study ◽  
Branching Processes ◽  
Random Variables ◽  
General Context ◽  
Growth Property ◽  
The Laplace Transform ◽  
Laplace Type ◽  
Made In

We study the exponential growth of branching processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells’s numbers are also assumed to be dependent. Applying the spectral study of Laplace-type operators recently made in Hervé et al. [ESAIM: PS 23 (2019) 607–637], we illustrate our results in the Markov context, for which the exponential growth property is linked to the Laplace transform of the lifetimes of the cells.

Continuity for multi-type branching processes with varying environments

1999 ◽  
Vol 36 (01) ◽  
pp. 139-145 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Linear Combination ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Main Tool ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (03) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

A proof of characterisation of oscillation for higher-order neutral differential equations of mixed type by the Laplace transform

1994 ◽  
Vol 124 (5) ◽  
pp. 909-916 ◽  
Author(s):  
O. Arino ◽  
M. A. El Attar
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
General Expression ◽  
Exponential Growth ◽  
Mixed Type ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Equations Of Mixed Type ◽  
The Laplace Transform ◽  
Image Position

Consider the general expression of such equations in the formwhere Ai, Bj, ∊ ℝ, δo = 0 dn/ 0, dn are n-derivatives, n ≧ l, the σj'S and δj,'s respectively, are ordered as an increasing family with possibly positive and negative terms. These are the deviating arguments. In this paper, we provide a proof of this result based on the use of the Laplace transform. Our method involves new results regarding the exponential growth of positive solutions for such equations.

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (3) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

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”.

scholarly journals Couplings for locally branching epidemic processes

2014 ◽  
Vol 51 (A) ◽  
pp. 43-56
Author(s):  
A. D. Barbour
Keyword(s):  
Asymptotic Behaviour ◽  
Exponential Growth ◽  
Branching Process ◽  
Branching Processes ◽  
Epidemic Models ◽  
Limit Theory ◽  
First Order ◽  
Early Stages ◽  
Extra Step ◽  
History Of

The asymptotic behaviour of many locally branching epidemic models can, at least to first order, be deduced from the limit theory of two branching processes. The first is Whittle's (1955) branching approximation to the early stages of the epidemic, the phase in which approximately exponential growth takes place. The second is the susceptibility approximation; the backward branching process that approximates the history of the contacts that would lead to an individual becoming infected. The simplest coupling arguments for demonstrating the closeness of these branching process approximations do not keep the processes identical for quite long enough. Thus, arguments showing that the differences are unimportant are also needed. In this paper we show that, for some models, couplings can be constructed that are sufficiently accurate for this extra step to be dispensed with.

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