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The Hàjek-Rènyi inequality for Banach space valued martingales and the p smoothness of Banach spaces

✍ Scribed by Gan Shixin

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
158 KB
Volume
32
Category
Article
ISSN
0167-7152

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✦ Synopsis

A p-smoothable Banach space is characterized in terms of the Hhjek-R~nyi inequality for Banach space valued martingales. As applications of this inequality, the strong law of large numbers and integrability of the supremum of Banach space valued martingales are also given.

Keywords." Banach space valued martingale; Hhjek-R~nyi inequality; p-smoothable space; Strong law of large numbers H~jek and R~nyi (1955) proved the following important inequality: If (D~, n >~ 1 ) is a sequence of independent real random variables with EDn = 0 and ED2n < ~, n/> 1 and (bn, n/> 1 ) is a positive nondecreasing real sequence, then for any g > 0, any positive integers m < n,

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