Moments of the maximum in a critical branching process

1984 ◽  
Vol 21 (4) ◽  
pp. 920-923 ◽  
Author(s):  
Howard Weiner
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Image Position ◽  
Number Of Cells ◽  
Critical Branching Process

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let Martingale arguments are used to show that for some 0<a≦b<∞

Moments of the maximum in a critical branching process

1984 ◽  
Vol 21 (04) ◽  
pp. 920-923 ◽  
Author(s):  
Howard Weiner
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Image Position ◽  
Number Of Cells ◽  
Critical Branching Process

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let Martingale arguments are used to show that for some 0&lt;a≦b&lt;∞

A branching-process solution of the polydisperse coagulation equation

1984 ◽  
Vol 16 (01) ◽  
pp. 56-69 ◽  
Author(s):  
John L. Spouge
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Critical Time ◽  
Coagulation Equation ◽  
Multitype Branching Processes ◽  
Irreversible Aggregation ◽  
Image Position ◽  
Supercritical Branching Processes ◽  
Critical Branching Process

The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when The critical time tc when diverges corresponds to a critical branching process, while post-critical times t&gt; tc correspond to supercritical branching processes.

A branching-process solution of the polydisperse coagulation equation

1984 ◽  
Vol 16 (1) ◽  
pp. 56-69 ◽  
Author(s):  
John L. Spouge
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Critical Time ◽  
Coagulation Equation ◽  
Multitype Branching Processes ◽  
Irreversible Aggregation ◽  
Image Position ◽  
Supercritical Branching Processes ◽  
Critical Branching Process

The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when The critical time tc when diverges corresponds to a critical branching process, while post-critical times t> tc correspond to supercritical branching 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.

Remarks on the maxima of a martingale sequence with application to the simple critical branching process

1987 ◽  
Vol 24 (03) ◽  
pp. 768-772 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Branching Process ◽  
Alternative Proof ◽  
Watson Process ◽  
Image Position ◽  
Critical Branching Process

Let where {Zn, ℱn } is a non-negative submartingale satisfying Ei (Zn log Zn ) →∞. It is shown that When {Zn } is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

Remarks on the maxima of a martingale sequence with application to the simple critical branching process

1987 ◽  
Vol 24 (3) ◽  
pp. 768-772 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Branching Process ◽  
Alternative Proof ◽  
Watson Process ◽  
Image Position ◽  
Critical Branching Process

Let where {Zn, ℱn} is a non-negative submartingale satisfying Ei(Zn log Zn) →∞. It is shown that When {Zn} is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

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.

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.

scholarly journals Period-Life of a Branching Process with Migration and Continuous Time

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 868
Author(s):  
Khrystyna Prysyazhnyk ◽  
Iryna Bazylevych ◽  
Ludmila Mitkova ◽  
Iryna Ivanochko
Keyword(s):  
Random Process ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Probability Generating Function ◽  
Time Interval ◽  
The Moment ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

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