Self-normalized large deviation for supercritical branching processes

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.

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Large deviations for supercritical multitype branching processes

Journal of Applied Probability ◽

10.1017/s0021900200020490 ◽

2004 ◽

Vol 41 (03) ◽

pp. 703-720 ◽

Cited By ~ 1

Author(s):

Owen Dafydd Jones

Keyword(s):

Large Deviations ◽

Growth Rates ◽

Branching Process ◽

Large Deviation ◽

Branching Processes ◽

Log P ◽

Single Individual ◽

Periodic Functions ◽

Multitype Branching Process ◽

Multitype Branching Processes

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x α F *[i](x) + o(x α ) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x −β/(1−β) G*[i](x) + o (x −β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = x δ/(δ−1) H *[i](x) + o(x δ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F *, G * and H * are multiplicatively periodic functions.

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A branching-process solution of the polydisperse coagulation equation

Advances in Applied Probability ◽

10.2307/1427224 ◽

1984 ◽

Vol 16 (1) ◽

pp. 56-69 ◽

Cited By ~ 4

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.

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Large deviations for supercritical multitype branching processes

Journal of Applied Probability ◽

10.1239/jap/1091543420 ◽

2004 ◽

Vol 41 (3) ◽

pp. 703-720 ◽

Cited By ~ 4

Author(s):

Owen Dafydd Jones

Keyword(s):

Large Deviations ◽

Growth Rates ◽

Branching Process ◽

Large Deviation ◽

Branching Processes ◽

Log P ◽

Single Individual ◽

Periodic Functions ◽

Multitype Branching Process ◽

Multitype Branching Processes

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + xαF*[i](x) + o(xα) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x−β/(1−β) G*[i](x) + o (x−β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = xδ/(δ−1)H*[i](x) + o(xδ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F*, G* and H* are multiplicatively periodic functions.

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Continuity for multi-type branching processes with varying environments

Journal of Applied Probability ◽

10.1017/s0021900200016910 ◽

1999 ◽

Vol 36 (01) ◽

pp. 139-145 ◽

Cited By ~ 1

Author(s):

Owen Dafydd Jones

Keyword(s):

Linear Combination ◽

Branching Process ◽

Branching Processes ◽

Random Variables ◽

Main Tool ◽

Concentration Function ◽

Independent Random Variables ◽

Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

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On the convergence of the supercritical branching processes with immigration

Journal of Applied Probability ◽

10.2307/3213010 ◽

1977 ◽

Vol 14 (2) ◽

pp. 387-390 ◽

Cited By ~ 3

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.

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Integer-valued branching processes with immigration

Advances in Applied Probability ◽

10.1017/s0001867800021571 ◽

1983 ◽

Vol 15 (04) ◽

pp. 713-725 ◽

Cited By ~ 1

Author(s):

F. W. Steutel ◽

W. Vervaat ◽

S. J. Wolfe

Keyword(s):

Difference Equation ◽

Continuous Time ◽

Queueing Theory ◽

Branching Processes ◽

Random Variables ◽

Discrete State ◽

Limiting Behaviour ◽

Supercritical Branching Processes

The notion of self-decomposability for -valued random variables as introduced by Steutel and van Harn [10] and its generalization by van Harn, Steutel and Vervaat [5], are used to study the limiting behaviour of continuous-time Markov branching processes with immigration. This behaviour provides analogues to the behaviour of sequences of random variables obeying a certain difference equation as studied by Vervaat [12] and their continuous-time counterpart considered by Wolfe [13]. An application in queueing theory is indicated. Furthermore, discrete-state analogues are given for results on stability in the processes studied by Wolfe, and for results on self-decomposability in supercritical branching processes by Yamazato [14].

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A limit theorem for a class of supercritical branching processes

Journal of Applied Probability ◽

10.1017/s002190020003610x ◽

1972 ◽

Vol 9 (04) ◽

pp. 707-724 ◽

Cited By ~ 17

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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Existence and positivity of the limit in processes with a branching structure

Journal of Applied Probability ◽

10.1239/jap/1032374235 ◽

1999 ◽

Vol 36 (1) ◽

pp. 132-138

Author(s):

M. P. Quine ◽

W. Szczotka

Keyword(s):

Stochastic Process ◽

Branching Process ◽

Random Variables ◽

Random Variable ◽

Natural Generalization ◽

Partial Sums ◽

Urn Model ◽

Special Case ◽

Independent And Identically Distributed

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

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Coalescence on critical and subcritical multitype branching processes

Journal of Applied Probability ◽

10.1017/jpr.2016.41 ◽

2016 ◽

Vol 53 (3) ◽

pp. 802-817

Author(s):

Jyy-I Hong

Keyword(s):

Limit Distribution ◽

Joint Distribution ◽

Branching Process ◽

Common Ancestor ◽

Branching Processes ◽

Recent Common Ancestor ◽

Most Recent Common Ancestor ◽

Generation Number ◽

Multitype Branching Processes ◽

Lines Of Descent

AbstractConsider a d-type (d<∞) Galton–Watson branching process, conditioned on the event that there are at least k≥2 individuals in the nth generation, pick k individuals at random from the nth generation and trace their lines of descent backward in time till they meet. In this paper, the limit behaviors of the distributions of the generation number of the most recent common ancestor of any k chosen individuals and of the whole population are studied for both critical and subcritical cases. Also, we investigate the limit distribution of the joint distribution of the generation number and their types.

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