Lithuanian Mathematical Journal

, Volume 59, Issue 4, pp 575–594 | Cite as

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

  • Rimas NorvaišaEmail author
  • Alfredas Račkauskas


Let X, X1, X2, . . . be a sequence of nondegenerate i.i.d. random variables, let μ = {μni : 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.


self-normalized CLT weak invariance principle random measure process p-variation convergence in distribution uniform central limit theorem 


primary 60F17 secondary 60G57 


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Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania

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