Markov processes associated with critical Galton-Watson processes with application to extinction probabilities

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

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On the parity of individuals in a branching process

Journal of Applied Probability ◽

10.1017/s0021900200094286 ◽

1976 ◽

Vol 13 (02) ◽

pp. 219-230 ◽

Cited By ~ 9

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.

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Some limit theorems for Markov branching processes

Journal of Applied Probability ◽

10.1017/s0021900200095309 ◽

1973 ◽

Vol 10 (02) ◽

pp. 299-306 ◽

Cited By ~ 1

Author(s):

J. R. Leslie

Keyword(s):

Limit Theorems ◽

Branching Process ◽

Branching Processes ◽

Original Process ◽

The Central Limit Theorem ◽

Iterated Logarithm ◽

Logarithm Law ◽

Increasing Process ◽

Watson Process ◽

Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

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Supercritical branching processes with density independent catastrophes

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065579 ◽

1988 ◽

Vol 104 (2) ◽

pp. 413-416 ◽

Cited By ~ 8

Author(s):

D. R. Grey

Keyword(s):

Continuous Time ◽

Branching Process ◽

Transition Probabilities ◽

Branching Processes ◽

Constant Rate ◽

Watson Process ◽

Stationary Transition ◽

Supercritical Branching Processes ◽

Independent Identically Distributed ◽

Independent Poisson Process

A Markov branching process in either discrete time (the Galton–Watson process) or continuous time is modified by the introduction of a process of catastrophes which remove some individuals (and, by implication, their descendants) from the population. The catastrophe process is independent of the reproduction mechanism and takes the form of a sequence of independent identically distributed non-negative integer-valued random variables. In the continuous time case, these catastrophes occur at the points of an independent Poisson process with constant rate. If at any time the size of a catastrophe is at least the current population size, then the population becomes extinct. Thus in both discrete and continuous time we still have a Markov chain with stationary transition probabilities and an absorbing state at zero. Some authors use the term ‘emigration’ as an alternative to ‘catastrophe’.

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Deviations for Jumping Times of a Branching Process Indexed by a Poisson Process

Mathematical Problems in Engineering ◽

10.1155/2019/6137926 ◽

2019 ◽

Vol 2019 ◽

pp. 1-7

Author(s):

Yanhua Zhang ◽

Zhenlong Gao

Keyword(s):

Poisson Process ◽

Continuous Time ◽

Branching Process ◽

Time Process ◽

Rate Of Growth ◽

Continuous Time Process ◽

Watson Process

Consider a continuous time process {Yt=ZNt, t≥0}, where {Zn} is a supercritical Galton–Watson process and {Nt} is a Poisson process which is independent of {Zn}. Let τn be the n-th jumping time of {Yt}, we obtain that the typical rate of growth for {τn} is n/λ, where λ is the intensity of {Nt}. Probabilities of deviations n-1τn-λ-1>δ are estimated for three types of positive δ.

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Skeletal stochastic differential equations for continuous-state branching processes

Journal of Applied Probability ◽

10.1017/jpr.2019.67 ◽

2019 ◽

Vol 56 (4) ◽

pp. 1122-1150 ◽

Cited By ~ 1

Author(s):

D. Fekete ◽

J. Fontbona ◽

A. E. Kyprianou

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Continuous Time ◽

Branching Process ◽

Branching Processes ◽

Continuous State ◽

Continuous State Branching Process ◽

Common Framework ◽

Watson Process ◽

Skeletal Representation

AbstractIt is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

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Some limit theorems for Markov branching processes

Journal of Applied Probability ◽

10.2307/3212347 ◽

1973 ◽

Vol 10 (2) ◽

pp. 299-306 ◽

Cited By ~ 3

Author(s):

J. R. Leslie

Keyword(s):

Limit Theorems ◽

Branching Process ◽

Branching Processes ◽

Original Process ◽

The Central Limit Theorem ◽

Iterated Logarithm ◽

Logarithm Law ◽

Increasing Process ◽

Watson Process ◽

Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

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Large deviations for Lotka-Nagaev estimator of a randomly indexed branching process

Filomat ◽

10.2298/fil1817803g ◽

2018 ◽

Vol 32 (17) ◽

pp. 5803-5808 ◽

Cited By ~ 1

Author(s):

Zhenlong Gao ◽

Lina Qiu

Keyword(s):

Large Deviations ◽

Renewal Process ◽

Continuous Time ◽

Branching Process ◽

Asymptotic Properties ◽

Time Process ◽

Continuous Time Process ◽

Watson Process

Consider a continuous time process {Yt=ZNt, t ? 0}, where {Zn} is a supercritical Galton-Watson process and {Nt} is a renewal process which is independent of {Zn}. Firstly, we study the asymptotic properties of the harmonic moments E(Y-rt) of order r > 0 as t ? ?. Then, we obtain the large deviations of the Lotka-Negaev estimator of offspring mean.

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On the parity of individuals in a branching process

Journal of Applied Probability ◽

10.2307/3212825 ◽

1976 ◽

Vol 13 (2) ◽

pp. 219-230 ◽

Cited By ~ 16

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.

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

Mathematics ◽

10.3390/math9080868 ◽

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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