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

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 η_nj’s are assumed to be defined on some basic probability space (p-space) $(\Omega,A,\mathbb{P})$ with values in X such that each η_nj : 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 1_C denotes the indicator function of C.