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Diffusion Approximation of an Array of Controlled Branching Processes

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Abstract

In this paper an array of controlled branching processes is considered. Using operator semigroup convergence theorems, it is proved that the fluctuation limit is a diffusion process under the conditions that the offspring and control means tend to be critical. As an application of this result, in a parametric framework, it is obtained that the bootstrapping distribution of the weighted conditional least squares estimator of the offspring mean in the critical case is not consistent. From this, it is concluded that the standard parametric bootstrap weighted conditional least squares estimate is asymptotically invalid in the critical case.

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Authors and Affiliations

  1. Department of Mathematics, University of Extremadura, Avda. Elvas s/n, 06006, Badajoz, Spain

    Miguel González & Inés M. del Puerto

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  1. Miguel González
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  2. Inés M. del Puerto
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Correspondence to Inés M. del Puerto.

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González, M., del Puerto, I.M. Diffusion Approximation of an Array of Controlled Branching Processes. Methodol Comput Appl Probab 14, 843–861 (2012). https://doi.org/10.1007/s11009-012-9285-8

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  • DOI: https://doi.org/10.1007/s11009-012-9285-8

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