Uniform asymptotic normality of self-normalized weighted sums of random variables*

Abstract

Let X, X1, X2, . . . be a sequence of nondegenerate i.i.d. random variables, let μ = {μni : n ∈ ℕ+, i = 1, …, n} be a triangular array of possibly random probabilities on the interval [0, 1], and let $$ \mathcal{F} $$ be a class of functions with bounded q-variation on [0, 1] for some q ∈ [1, 2). We prove the asymptotic normality uniformly over $$ \mathcal{F} $$ of self-normalized weighted sums $$ {\sum}_{i=1}^n{X}_i{\mu}_{ni} $$ when μ is the array of point measures, uniform probabilities, and their random versions. Also, we prove a weak invariance principle in the Banach space of functions of bounded p-variation with p > 2 for partial-sum processes, polygonal processes, and their adaptive versions.