Random Sums and Branching Stochastic Processes

Rahimov, I.

doi:10.1007/978-1-4612-4216-1

Cited by 44 publications

(21 citation statements)

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“…and appealing to Lemma 4, we find that the second term in (16) converges in probability to 0 as n tends to ∞. Hence, condition (14) is satisfied with C(t) = t 1+β .…”

Section: Subsequent Application Of This Identity Leads To the Relationmentioning

confidence: 76%

Functional limit theorems for critical processes with immigration

Rahimov

2007

Adv. Appl. Probab.

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We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is n √ α(n), where α(n) denotes the mean number of immigrating individuals in the nth generation.

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“…and appealing to Lemma 4, we find that the second term in (16) converges in probability to 0 as n tends to ∞. Hence, condition (14) is satisfied with C(t) = t 1+β .…”

Section: Subsequent Application Of This Identity Leads To the Relationmentioning

confidence: 76%

Functional limit theorems for critical processes with immigration

Rahimov

2007

Adv. Appl. Probab.

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“…The theory of branching processes has a long and venerable history [1,2,11,14,20,21,22,32,38]. Connections with specific applications, such as surname survival and mutant gene extinction, were initially so strong that the basic concepts were rediscovered in several different settings.…”

Section: Introductionmentioning

confidence: 99%

In the Garden of Branching Processes

2004

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The current paper surveys and develops numerical methods for Markovian multitype branching processes in continuous time. Particular attention is paid to the calculation of means, variances, extinction probabilities, and marginal distributions in the presence of a Poisson stream of immigrant particles. The Poisson process assumption allows for temporally complex patterns of immigration and facilitates application of marked Poisson processes and Campbell's formulas. The methods and formulas derived are applied to four models: two population genetics models, a model for vaccination against an infectious disease in a community of households, and a model for the growth of resistant HIV virus in patients undergoing drug therapy.

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“…Investigations show that the asymptotic behavior of the process with immigration is very sensitive to any changes of the immigration process in time. For instance, in the critical case, a change in the mean number of immigrating individuals in time leads to such fluctuations of the Functional limit theorems for critical processes 1055 process that it is necessary to use various functional normalizations to obtain a nondegenerate limit distribution for the process; see [16,Chapter III] and the references therein. Therefore, a description of the processes that can be used as an approximation in this situation is of interest.…”

Section: Introductionmentioning

confidence: 99%

Functional limit theorems for critical processes with immigration

Rahimov

2007

Advances in Applied Probability

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We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

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Random Sums and Branching Stochastic Processes

Abstract: In what follows all references to monographs are applicable also to multiauthorship volumes such as seminar notes.

Cited by 44 publications

References 0 publications

Functional limit theorems for critical processes with immigration

Functional limit theorems for critical processes with immigration

In the Garden of Branching Processes

Functional limit theorems for critical processes with immigration

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