ON ASYMPTOTIC RELATIONS FOR THE CRITICAL GALTON – WATSON PROCESS

2021 ◽  
pp. 5-16
Author(s):  
K.E. Kudratov ◽  
◽  
Y.M. Khusanbaev ◽  
Keyword(s):  
Single Particle ◽  
Branching Process ◽  
Critical Case ◽  
Random Number ◽  
Branching Processes ◽  
Factorial Moment ◽  
Subcritical Case ◽  
Watson Process ◽  
Third Moment ◽  
Number Of Particles

Determining the asymptotics of the continuation probability for a Galton–Watson branching process is one of the most important problems in the theory of branching processes. This problem was solved by A.N. Kolmogorov (1938) in the case when the process starts with a single particle, and the classical result is obtained. A similar result for continuous branching processes was proved by B.A. Sevastyanov (1951). The next term in the expansion for continuous branching processes was obtained by V.M. Zolotarev (1957). The next term in the expansion for continuous branching processes in the critical case was obtained by V.P. Chistyakov (1957); the asymptotic expansion in the subcritical case under the condition of finiteness of the k-factorial moment was obtained by R. Mukhamedkhanova (1966). Asymptotic expansions for discrete branching processes in the subcritical and supercritical cases, provided that any m-factorial moment is finite, were obtained by S.V. Nagaev and R. Mukhamedkhanova (1966). In the critical case, the weak convergence of the conditional distribution of the quantity P(Z(n) > 0)Z(n) under the condition Z(n) > 0 to the exponential distribution was proved by A.M. Yaglom (1947) for processes starting with a single particle in the case of finiteness of the third moment of the number of generations. Subsequently, Spitzer, Kesten, and Ney (1966) proved this result under the condition that the second moment is finite. A similar result for branching processes with continuous parameters was established by V.M. Zolotarev (1957). In this paper, we study the asymptotics of the probability of continuation of the critical Galton-Watson process, starting with η particles. In addition, we prove an analogue of Yaglom’s theorem for critical Galton – Watson processes starting with a random number of particles.

scholarly journals Multi-type subcritical branching processes in a random environment

2018 ◽  
Vol 50 (A) ◽  
pp. 281-289 ◽  
Author(s):  
Vladimir Vatutin ◽  
Vitali Wachtel
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Subcritical Case ◽  
One Dimensional ◽  
Conditional Limit Theorem ◽  
Long Time ◽  
The One ◽  
Conditional Limit ◽  
Number Of Particles

Abstract We study the asymptotic behavior of the survival probability of a multi-type branching process in a random environment. In the one-dimensional situation, the class of processes considered corresponds to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process given its survival for a long time.

scholarly journals Skeletal stochastic differential equations for superprocesses

2020 ◽  
Vol 57 (4) ◽  
pp. 1111-1134
Author(s):  
Dorottya Fekete ◽  
Joaquin Fontbona ◽  
Andreas E. Kyprianou
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Subcritical Case ◽  
Markov Branching Process ◽  
Continuous State ◽  
Common Framework ◽  
Infinite Line ◽  
Current Article ◽  
Lines Of Descent ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

1976 ◽  
Vol 13 (3) ◽  
pp. 455-465
Author(s):  
D. I. Saunders
Keyword(s):  
State Space ◽  
Branching Process ◽  
Branching Processes ◽  
Rates Of Convergence ◽  
Expected Number ◽  
Subcritical Case ◽  
Arbitrary State ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

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 &gt; 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 note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (03) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 &lt; m = h'(1 –) &lt; ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) &lt; 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

A limit theorem for a class of supercritical branching processes

1972 ◽  
Vol 9 (04) ◽  
pp. 707-724 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Generating Functional ◽  
Number Of Children ◽  
Supercritical Case ◽  
Age Dependent ◽  
Watson Process ◽  
Supercritical Branching Processes ◽  
Convergence In Mean

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn } form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ 1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn /mn to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].

scholarly journals Fluctuation limit of branching processes with immigration and estimation of the means

2005 ◽  
Vol 37 (02) ◽  
pp. 523-538 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Critical Case ◽  
Branching Processes ◽  
Critical Value ◽  
Subcritical Case ◽  
Asymptotically Normal ◽  
Fluctuation Limit ◽  
Conditional Least Squares ◽  
Type Process ◽  
Conditional Least Squares Estimator

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2, in contrast to both the subcritical case, in which it is n 1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

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.

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