An absolute moments condition for normed sums of independent variables
type: Item
instance of: scholarly article
title: An absolute moments condition for normed sums of independent variables
zbMATH Open document ID: 0701.60015
DOI: 10.1007/BF01056488
published in: Ukrainian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let \(X_ n\), \(n\in N\), be i.i.d. symmetric random variables, \(1\leq p<q<2\). It is shown that the relation \[ 0<\inf | n^{- 1/q}\sum^{n}_{k=1}X_ k| \leq \sup | n^{- 1/q}\sum^{n}_{k=1}X_ k| <\infty \] is satisfied iff E \(X_ 1=0\) and there are constants \(a,b,c>0\) such that for \(x\geq c\) \[ ax^{-q}\leq P[| X_ 1| \geq x]\leq bx^{-q}. \] The case \(0<p<q<2\), \(p<2\), is also considered. The obtained assertions are refinements of the results of \textit{C. G. Esseen} and \textit{S. Janson}, Stochastic Processes Appl. 19, 173-182 (1985; Zbl 0554.60050).
Mathematics Subject Classification ID: 60E99
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4149247
zbMATH Keywords: characteristic function
zbMATH Keywords: symmetric random variables
author: Michael Braverman
reviewed by: V. Tarieladze
MaRDI profile type: Publication
cites work: On moment conditions for normed sums of independent variables and martingale differences
cites work: Q3689920
cites work: Q5644834
full work available at URL: https://doi.org/10.1007/bf01056488
OpenAlex ID: W2095745428
description: scientific article; zbMATH DE number 4149247