Approximation of a sum of martingale differences generated by a bootstrap branching process

Ibrahim Rahimov

Document Type

Book Chapter

Source of Publication

Workshop on Branching Processes and Their Applications

Publication Date

1-22-2010

Abstract

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{Z(k), k\geq 0\}$$\end{document} be a branching stochastic process with non-stationary immigration given by offspring distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p_{j}(\theta),j\geq 0\}$$\end{document} depending on unknown parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta\in \Theta$$\end{document}. We estimate θ by an estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta}_{n}$$\end{document} based on sample \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X}_{n}=\{Z(i), i=1, {\ldots}, n\}$$\end{document}. Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X}_{n}$$\end{document}, we generate bootstrap branching process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{Z^{\mathcal{X}_{n}}(k), k\geq 0\}$$\end{document} for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n=1, 2, {\ldots}$$\end{document} with offspring distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p_{j}(\hat{\theta}_{n}), j\geq 0\}$$\end{document}. In the paper we address the following question: How good must be estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta}_{n}$$\end{document}, the bootstrap process to have the same asymptotic properties as the original process? We obtain conditions for the estimator which are sufficient and necessary for this in critical case. To derive these conditions we investigate a weighted sum of martingale differences generated by an array of branching processes. We provide a general functional limit theorem for this sum, which includes critical or nearly critical processes with increasing or stationary immigration and with large or fixed number of initial ancestors. It also includes processes without immigration with increasing random number of initial individuals. Possible applications in estimation theory of branching processes are also be provided.

ISSN

0930-0325

Publisher

Springer Berlin Heidelberg

Volume

197

First Page

121

Last Page

133

Disciplines

Physical Sciences and Mathematics

Indexed in Scopus

no

Open Access

no

Link to Full Text

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