A Uniform Law of Large Numbers for Set-Indexed Processes with Applications to Empirical and Partial-Sum Processes

  • Peter Gaenssler
  • Klaus Ziegler
Conference paper
Part of the Progress in Probability book series (PRPR, volume 35)

Abstract

The purpose of the present paper is to establish a uniform law of large numbers (ULLN) in form of a Mean Glivenko-Cantelli result for so-called partial-sum processes with random locations and indexed by Vapnik-Chervonenkis classes (VCC) of sets in arbitrary sample spaces. The context is as follows: Let X = (X, x) be an arbitrary measurable space, \((\eta_{nj})_{1 \leq j \leq j (n),n \in \mathbb{N}}\) be a triangular array of random elements (r.e.) in X (that is, the ηnj’s are assumed to be defined on some basic probability space (p-space) \((\Omega,A,\mathbb{P})\) with values in X such that each ηnj : Ω → X is A \(\mathfrak{X}\)-measurable), and let \((\xi _{nj})_{1 \leq j \leq j(n), n \in \mathbb{N}}\) be a triangular array of real-valued random variables (r.v.) (also defined on \((\Omega,A,\mathbb{P}))\) such that for each \(n\in \mathbb{N} (\eta_{n1},\xi_{n1}),\cdots, (\eta_{nj(n)},\xi_{nj(n)})\) is a sequence of independent but not necessarily identically distributed (i.d.) pairs of r.e.’s in \((X \times \mathbb{R},X \otimes \mathbb{B})\), where \(X \otimes \mathbb{B}\) denotes the product σ-field of x and the Borel σ-field IB in ℝ; i.e. the components within each pair need not be independent. Given a class \(C\subset X\), define a set-indexed process (of sample size \(n \in \mathbb{N})S_n =(S_n(C))_{C\in C}\) by
$$S_n(C):= \sum_{j \geq (n)}\;\;\;\;1_C(\eta_{nj})\xi_{nj},\;\;\;\;\;C \in C,\;\;\;\;\;\;\;\;\;\;(1.1)$$
where 1C denotes the indicator function of C.

Keywords

Entropy Zinn 

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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Peter Gaenssler
    • 1
  • Klaus Ziegler
    • 1
  1. 1.Math. InstituteUniv. of MunichMunichGermany

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