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On Recent Developments in the Theory of Set-Indexed Processes

A unified approach to empirical and partial-sum processes

  • Conference paper
Asymptotic Statistics

Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

Abstract

Two important processes in probability and statistics are the empirical and partial-sum processes. The purpose of this paper is to présenta unified approach to both types of processes including their multi-variate versions by studying processes Sn=((Sn(C))C ∈ with Sn(C) being defined by

$$ {S_n}(C): = \sum\limits_{j\underline < j(n)} {{1_C}({\eta _{nj}}){\xi _{nj}},C \in \ell ,} $$

the ηn j’s being random elements (random locations) in an arbitrary sample space X, the ξ’n js being real-valued random variables (random masses), and where the index set is a Vapnik-Chervonenkis class of subsets of X.

Our emphasis is on a asymptotic results (as the sample size n tends to infinity) for the processes Sn such as functional central limit theorems (FCLT) and a uniform law of large numbers (ULLN) (the uniformity being w.r.t.), containing various results for empirical and partial-sum processes as special cases.

Invited Lecture for the Fifth Prague Symposium on Asymtotic Statistics, 4.-9. Sept., 1993

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Authors and Affiliations

  1. Math. Institute, University of Munich, D-80333, Munich, Deutschland

    Peter Gaenssler

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  1. Peter Gaenssler
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Editor information

Editors and Affiliations

  1. Department of Probability and Mathematical Statistics, Charles University, Sokolovská 83, CZ-186 00, Praha 8, Czech Republic

    Petr Mandl  & Marie Hušková  & 

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© 1994 Springer-Verlag Berlin Heidelberg

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Gaenssler, P. (1994). On Recent Developments in the Theory of Set-Indexed Processes. In: Mandl, P., Hušková, M. (eds) Asymptotic Statistics. Contributions to Statistics. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-57984-4_7

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  • DOI: https://doi.org/10.1007/978-3-642-57984-4_7

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-7908-0770-7

  • Online ISBN: 978-3-642-57984-4

  • eBook Packages: Springer Book Archive

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