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.
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* The research supported by the Research Council of Lithuania, grant No. S-MIP-17-76.
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Norvaiša, R., Račkauskas, A. Uniform asymptotic normality of self-normalized weighted sums of random variables*. Lith Math J 59, 575–594 (2019). https://doi.org/10.1007/s10986-019-09461-w
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DOI: https://doi.org/10.1007/s10986-019-09461-w