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)


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.


Random Location Empirical Process Maximal Inequality Triangular Array Empirical Characteristic Function 
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 1994

Authors and Affiliations

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

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