On probabilities of large deviations in Banach spaces
type: Item
instance of: scholarly article
title: On probabilities of large deviations in Banach spaces
zbMATH Open document ID: 0678.60005
DOI: 10.1007/BF01474639
publication date: 1990-01-01T00:00:00Z
review text: Let \(X,X_ 1,X_ 2,...\in B\) denote a sequence of i.i.d. random variables on a real separable Banach space B, \(Y\in B\) denotes a Gaussian random variable. Suppose that E X\(=E Y=0\) and that the covariances of X and Y coincide. Denote \(S_ n=n^{-1/2}(X_ 1+...+X_ n)\). We prove that under appropriate conditions \[ P(\| S_ n\| >r)=P(\| Y\| >r)(1+o(1))\quad as\quad n\to \infty \] and give estimates of the remainder term. Applications to the \(\omega^ 2\)-test, the Anderson- Darling test and to empirical characteristic functions are given.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 62G10
zbMATH DE Number: 4111668
zbMATH Keywords: omega-square tests
zbMATH Keywords: Linnik zone
zbMATH Keywords: Bernstein inequalities
zbMATH Keywords: Anderson-Darling test
zbMATH Keywords: empirical characteristic functions
author: Vidmantas Bentkus
author: Alfredas Račkauskas
MaRDI profile type: Publication
cites work: Q3906818
cites work: Estimates of the proximity of sums of independent random elements in the space \(C[0,1]\).
cites work: Lower bounds for the rate of convergence in the central limit theorem in Banach spaces
cites work: Q3761357
cites work: Estimates of the rate of approximation of independent random variables in a Banach space. I
cites work: Estimates of the rate of convergence of sums of independent random variables in a Banach space. II
cites work: Q3486652
cites work: Q5808639
cites work: Q5560061
cites work: Exact convergence rates in some martingale central limit theorems
cites work: On the rate of approximation in the central limit theorem
cites work: Q3660611
cites work: On the rate of convergence in the central limit theorem in Banach spaces
cites work: A note on the rate of convergence in the martingale central limit theorem
cites work: Q5535438
cites work: Berry-Esseen estimates in Hilbert space and an application to the law of the iterated logarithm
cites work: On the Rate of Convergence in the Central Limit Theorem in Certain Banach Spaces
cites work: Q3931833
cites work: Q4692961
cites work: Q4085017
cites work: Bad rates of convergence for the central limit theorem in Hilbert space
cites work: Large deviations of sums of independent random variables
cites work: Normal approximation - some recent advances
cites work: The Density of the Distribution of the Maximum of a Gaussian Process
cites work: Exponential inequalities for sums of random vectors
cites work: Approximation of Distributions of Sums of Independent Random Variables with Values in Infinite-Dimensional Spaces
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 4111668