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Uniform asymptotic normality of self-normalized weighted sums of random variables*
Lithuanian Mathematical Journal ( IF 0.4 ) Pub Date : 2019-10-01 , DOI: 10.1007/s10986-019-09461-w
Rimas Norvaiša , Alfredas Račkauskas

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.

中文翻译:

随机变量的自归一化加权和的均匀渐近正态性*

设 X, X1, X2, . . . 是一个非退化 iid 随机变量序列,令 μ = {μni : n ∈ ℕ+, i = 1, …, n} 是区间 [0, 1] 上可能随机概率的三角数组,并令 \( \ mathcal{F} \) 是一类在 [0, 1] 上有界 q 变化的函数,对于某些 q ∈ [1, 2)。我们证明了在 \( \mathcal{F} \) 自归一化加权和 \( {\sum}_{i=1}^n{X}_i{\mu}_{ni} \) 上一致的渐近正态性当 μ 是点度量、均匀概率及其随机版本的数组时。此外,我们在部分和过程、多边形过程及其自适应版本中证明了 p > 2 的有界 p 变化函数的 Banach 空间中的弱不变性原理。
更新日期:2019-10-01
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