On the parity of individuals in a branching process

1976 ◽  
Vol 13 (2) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (02) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

Linear growth in near-critical population-size-dependent multitype Galton–Watson processes

1989 ◽  
Vol 26 (3) ◽  
pp. 431-445 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Limit Distribution ◽  
Branching Process ◽  
Linear Growth ◽  
Left Eigenvector ◽  
Size Dependent ◽  
Second Moments ◽  
The Mean ◽  
Critical Process

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.

Linear growth in near-critical population-size-dependent multitype Galton–Watson processes

1989 ◽  
Vol 26 (03) ◽  
pp. 431-445
Author(s):  
Fima C. Klebaner
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Limit Distribution ◽  
Branching Process ◽  
Linear Growth ◽  
Left Eigenvector ◽  
Size Dependent ◽  
Second Moments ◽  
The Mean ◽  
Critical Process

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.

A maximum sequence in a critical multitype branching process

1991 ◽  
Vol 28 (04) ◽  
pp. 893-897
Author(s):  
Aurel Spåtaru
Keyword(s):  
Spectral Radius ◽  
Branching Process ◽  
Multitype Branching Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
The Right ◽  
Maximum Sequence

Let (Zn ) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn ) consider the right eigenvector u = (u 1, · ··, up ) > 0, and set . It is shown that lim inf, lim sup whenever Z 0 = i, where .

A Galton–Watson process with a threshold

2016 ◽  
Vol 53 (2) ◽  
pp. 614-621
Author(s):  
K. B. Athreya ◽  
H.-J. Schuh
Keyword(s):  
Positive Integer ◽  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Mean Value ◽  
Size Dependent ◽  
The Mean ◽  
Watson Process ◽  
Critical Branching Process

Abstract In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

scholarly journals Estimation in branching processes with restricted observations

2006 ◽  
Vol 38 (4) ◽  
pp. 1098-1115 ◽  
Author(s):  
Ronald Meester ◽  
Pieter Trapman
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Infected Individual ◽  
Classical Swine Fever ◽  
Lattice Parameters ◽  
Time Interval ◽  
The Mean ◽  
Watson Process

We consider an epidemic model where the spread of the epidemic can be described by a discrete-time Galton-Watson branching process. Between times n and n + 1, any infected individual is detected with unknown probability π and the numbers of these detected individuals are the only observations we have. Detected individuals produce a reduced number of offspring in the time interval of detection, and no offspring at all thereafter. If only the generation sizes of a Galton-Watson process are observed, it is known that one can only estimate the first two moments of the offspring distribution consistently on the explosion set of the process (and, apart from some lattice parameters, no parameters that are not determined by those moments). Somewhat surprisingly, in our context, where we observe a binomially distributed subset of each generation, we are able to estimate three functions of the parameters consistently. In concrete situations, this often enables us to estimate π consistently, as well as the mean number of offspring. We apply the estimators to data for a real epidemic of classical swine fever.

A maximum sequence in a critical multitype branching process

1991 ◽  
Vol 28 (4) ◽  
pp. 893-897 ◽  
Author(s):  
Aurel Spåtaru
Keyword(s):  
Spectral Radius ◽  
Branching Process ◽  
Multitype Branching Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
The Right ◽  
Maximum Sequence

Let (Zn) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn) consider the right eigenvector u = (u1, · ··, up) > 0, and set . It is shown that lim inf, lim sup whenever Z0 = i, where .

Further approaches to computing fundamental characteristics of birth-death processes

2001 ◽  
Vol 38 (4) ◽  
pp. 995-1005 ◽  
Author(s):  
Frank Ball ◽  
Valeri T. Stefanov
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Exponential Family ◽  
Death Process ◽  
Random Sum ◽  
First Passage ◽  
Reward Functions ◽  
Passage Times ◽  
Birth Death

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

Further approaches to computing fundamental characteristics of birth-death processes

2001 ◽  
Vol 38 (04) ◽  
pp. 995-1005 ◽  
Author(s):  
Frank Ball ◽  
Valeri T. Stefanov
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Exponential Family ◽  
Death Process ◽  
Random Sum ◽  
First Passage ◽  
Reward Functions ◽  
Passage Times ◽  
Birth Death

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

scholarly journals A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process

2021 ◽  
Author(s):  
Michel Mandjes ◽  
Birgit Sollie
Keyword(s):  
Continuous Time ◽  
Real Life ◽  
Numerical Approach ◽  
Time Dependent ◽  
Death Process ◽  
Continuous Time Markov Chain ◽  
Likelihood Estimator ◽  
Real Life Data ◽  
The Matrix ◽  
Birth Death

AbstractThis paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

Sign in / Sign up

Export Citation Format

Share Document