Branching Processes with Immigration and Related Limit Theorems

Kawazu, Kiyoshi; Watanabe, Satoshi

doi:10.1137/1116003

Cited by 258 publications

(259 citation statements)

References 14 publications

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“…The motivation comes not only from the apparently convincing fitting found by using various graphical methods and several data sets, but also from the existing realistic models for the DLG processes which are the component processes for our superposition step. In particular, the DLG process is a CBI process with a.s. continuous trajectories, see Theorem 1, and the latter process can be represented as a scaled limit of Galton-Watson processes with immigration, see Theorem 2.2 in [20]. Another practical model is a Cox process with DLG process intensity, resulting in a birth and death process with immigration, see [7].…”

Section: Open Problems and Directions Of Future Researchmentioning

confidence: 99%

Superposition of Diffusions with Linear Generator and its Multifractal Limit Process

Iglói

Terdik

2003

ESAIM: PS

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Section: Open Problems and Directions Of Future Researchmentioning

confidence: 99%

Superposition of Diffusions with Linear Generator and its Multifractal Limit Process

Iglói

Terdik

2003

ESAIM: PS

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“…This interest comes from the fact that these new models arise as limits of discrete population models where there are interactions between individuals or where the offspring distribution depends on the environment (see for instance Lambert [26], Kawasu and Watanabe [22], Bansaye and Simatos [4] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Branching Processes in a Lévy Random Environment

Palau

Pardo

2017

Acta Appl Math

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In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813-857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1-13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.

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“…Note that the convergence of finite-dimensional distributions of a sequence of branching processes with immigration has been investigated by Kawazu and Watanabe [10] and Aliev [1].…”

Section: Introductionmentioning

confidence: 99%

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

Ispány¹,

Pap²,

Zuijlen³

2005

Adv. Appl. Probab.

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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.

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Branching Processes with Immigration and Related Limit Theorems

Cited by 258 publications

References 14 publications

Superposition of Diffusions with Linear Generator and its Multifractal Limit Process

Superposition of Diffusions with Linear Generator and its Multifractal Limit Process

Branching Processes in a Lévy Random Environment

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

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