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.

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> tc correspond to supercritical branching processes.

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<a≦b<∞

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<∞

Self-normalized large deviation for supercritical branching processes

2018 ◽  
Vol 55 (2) ◽  
pp. 450-458
Author(s):  
Weijuan Chu
Keyword(s):  
Branching Process ◽  
Large Deviation ◽  
Branching Processes ◽  
Random Variables ◽  
The Self ◽  
Offspring Distribution ◽  
Multitype Branching Processes ◽  
Supercritical Branching Processes ◽  
Independent And Identically Distributed

Abstract We consider a supercritical branching process (Zn, n ≥ 0) with offspring distribution (pk, k ≥ 0) satisfying p0 = 0 and p1 > 0. By applying the self-normalized large deviation of Shao (1997) for independent and identically distributed random variables, we obtain the self-normalized large deviation for supercritical branching processes, which is the self-normalized version of the result obtained by Athreya (1994). The self-normalized large deviation can also be generalized to supercritical multitype branching processes.

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 convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (2) ◽  
pp. 387-390 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Limit Distribution Function ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

scholarly journals On estimation of the variances for critical branching processes with immigration

2010 ◽  
Vol 47 (02) ◽  
pp. 526-542
Author(s):  
Chunhua Ma ◽  
Longmin Wang
Keyword(s):  
Least Squares ◽  
Branching Process ◽  
Branching Processes ◽  
Limiting Distributions ◽  
Least Squares Estimators ◽  
Conditional Least Squares ◽  
Regularly Varying ◽  
Branching Process With Immigration ◽  
Critical Branching Process

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 &lt; α &lt; 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

Asymptotic behavior of near-critical multitype branching processes

1991 ◽  
Vol 28 (03) ◽  
pp. 512-519 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Discrete Time ◽  
Growth Rates ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Limiting Distribution ◽  
Size Dependent ◽  
Generalized Gamma ◽  
Multitype Branching Processes

Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.

Limit theorems for stochastic growth models. II

1972 ◽  
Vol 4 (3) ◽  
pp. 393-428 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Limit Theorems ◽  
Growth Models ◽  
Branching Processes ◽  
Random Variable ◽  
Present Situation ◽  
Stochastic Growth ◽  
Stochastic Growth Models ◽  
Zygotic Selection ◽  
Image Position ◽  
Supercritical Branching Processes

We consider d-dimensional stochastic processes which take values in (R+)d These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some Here τ:(R+)d→R +, |x| = σ1d |x(i)|, A {x ∈(R+)d: |x| 1} and T: A→A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Zn|ρnwp. This result is a generalization of the main limit theorem for supercritical branching processes; note, however, that in the present situation both ρ and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

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].

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