A necessary condition for the convergence of the isotrope discrepancy

Abstract

Given a sequence (ξ_i)_i∈N of independent identically distributed (i.i.d.) ℝ^k-valued random vectors with distribution $\mu = Q_{\xi _1 }$, the isotrope discrepancy D ^μ_n (ω) is defined by

, where μ ^ω_n denotes the empirical probability distribution and the supremum is taken over the class ℓ_k of all convex measurable subsets of ℝ^k. In the present paper it is proved that μ_c(e(C))=0 for all C ε ℓ_k whenever D ^μ_n (ω)→0 a.s., and where e(C) denotes the set of all extreme points of C.