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On Functional Limit Theorems for a Class of Stochastic Processes Indexed by Pseudo-Metric Parameter Spaces (with applications to empirical processes)

  • Peter Gaenssler
  • Wilhelm Schneemeier
Part of the Progress in Probability book series (PRPR, volume 20)

Abstract

Let T = (T, d) be a pseudo-metric space assumed to be totally bounded for the pseudo-metric d. Let \({{\ell }^{\infty }}\) (T) be the space of all bounded real valued functions on T equipped with the supremum norm \(\left\| \cdot \right\|T\) (defined by \(\left\| x \right\|T:=\sup \{|x(t)|:t\in T\},x\in {{\ell }^{\infty }}(T)\) ) and let S 0:= U b (T,d) be the subspace of \({{\ell }^{\infty }}\) (T) consisting of all uniformly d-continuous functions on T; note that S 0 is separable and closed in \(({{\ell }^{\infty }}(T),\left\| \cdot \right\|t)\) cf. Corollary 2 in Section 2 below.

Keywords

Random Element Polish Space Empirical Process Nonnegative Real Number Functional Limit Theorem 
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Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • Peter Gaenssler
    • 1
  • Wilhelm Schneemeier
    • 1
  1. 1.Mathematical InstituteUniversity of MunichMunich 2W.-Germany

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