The measure of a translated ball in uniformly convex spaces
type: Item
instance of: scholarly article
title: The measure of a translated ball in uniformly convex spaces
zbMATH Open document ID: 0711.60006
DOI: 10.1007/BF01046095
author: Michał Ryznar
author: Tomasz Żak
published in: Journal of Theoretical Probability
publication date: 1990-01-01T00:00:00Z
review text: Let E be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let \(\mu\) be the convolution of the distribution of a random series in E with independent, one-dimensional components and an arbitrary probability measure on E. Under some assumptions on the components and the smoothness of the norm, the authors show that there exists a constant c such that for every \(r\in E\) and \(t\geq 0,\) \[ | \mu (B_ t)-\mu (B_ t+r)| \leq c\| r\|^ q, \] where \(B_ t\) is an open ball with radius t and q depends on the properties of the norm. This is an extension of an earlier result of the second author for symmetric Gaussian measures on separable Hilbert spaces [Probab. Math. Stat. 10, No.2, 257-270 (1989)]. As an application of the above result the authors investigate the obtained estimate for symmetric Gaussian and stable measures on E.
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 4169734
zbMATH Keywords: uniformly convex Banach space
zbMATH Keywords: modulus of uniform convexity
zbMATH Keywords: smoothness of the norm
zbMATH Keywords: symmetric Gaussian measures
zbMATH Keywords: stable measures
MaRDI profile type: Publication
cites work: Q5519417
cites work: On the lower tail of Gaussian seminorms
cites work: Convergence to a stable distribution via order statistics
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cites work: Q4188284
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full work available at URL: https://doi.org/10.1007/bf01046095
OpenAlex ID: W2033662974
description: scientific article; zbMATH DE number 4169734