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Partial-Sum Processes with Random Locations and Indexed by Vapnik-Červonenkis Classes of Sets in Arbitrary Sample Spaces

  • Miguel A. Arcones
  • Peter Gaenssler
  • Klaus Ziegler
Chapter
Part of the Progress in Probability book series (PRPR, volume 30)

Abstract

The purpose of the present paper is to establish a functional central limit theorem (FCLT) for partial-sum processes with random locations and indexed by Vapnik-Cervonenkis classes (VCC) of sets in arbitrary sample spaces. The context is as follows: Let X = (X,X) be an arbitrary measurable space, (ηj)j∈N be a sequence of independent and identically distributed (i.i.d.) random elements (r.e.) in X with distribution v on X (that is, the η; j ’s are asumed to be defined on some basic probability space (Ω, F, P) with values in X such that each ηj: (Ω, F) → (X,X) is measurable), and let (ξnj)1≤ j j (n),n∈ℕ be a triangular array of rowwise independent (but not necessarily identically distributed) real-valued random variables (r.v.) (also defined on (Ω, F, P)) such that the whole triangular arrray is independent of the sequence (ηj)j∈ℕ. Given a class CX, define a partial-sum process (of sample size nIN) S n = (S n(C))C∈c by
$$S_{n}:=\sum_{j\leq j(n)}I_{C}(\eta_{j})\xi_{nj}, C\in c$$
where I C denotes the indicator function of C.

Keywords

Central Limit Theorem Gaussian Process Random Location Maximal Inequality Functional Central Limit Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Miguel A. Arcones
    • 1
  • Peter Gaenssler
    • 2
  • Klaus Ziegler
    • 2
  1. 1.Math. Sci. Res. Inst.BerkeleyUSA
  2. 2.Math. InstituteUniv. of MunichMunich 2Germany

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