Characterization of Hilbert spaces by the strong law of large numbers
type: Item
instance of: scholarly article
title: Characterization of Hilbert spaces by the strong law of large numbers
zbMATH Open document ID: 0606.60009
DOI: 10.1016/0047-259X(86)90025-4
author: Jun Kawabe
published in: Journal of Multivariate Analysis
publication date: 1986-01-01T00:00:00Z
review text: Let X be a separable Banach space, \((\xi_ n)_{n\in N}\) be a sequence of independent strong second order mean zero random elements in X. It is shown that the convergence of the series \(\sum n^{-2}S_{\xi_ n}\) \((S_{\xi_ n}\) is the covariance operator of \(\xi_ n)\) in the space of nuclear operators implies \(n^{-1}\| \sum^{n}_{k=1}\xi_ k\| \to 0\) a.s. if and only if X is isomorphic to a Hilbert space.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60B11
zbMATH DE Number: 3980112
zbMATH Keywords: law of large numbers
zbMATH Keywords: relative compactness
zbMATH Keywords: Banach space type
zbMATH Keywords: covariance operator
zbMATH Keywords: nuclear operators
reviewed by: V. Tarieladze
MaRDI profile type: Publication
cites work: Q5341297
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description: scientific article; zbMATH DE number 3980112