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

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 C ⊂ X, define a partial-sum process (of sample size n ∈ IN) 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.