An infinite-dimensional law of large numbers in Cesaro's sense
type: Item
instance of: scholarly article
title: An infinite-dimensional law of large numbers in Cesaro's sense
zbMATH Open document ID: 0707.60011
DOI: 10.1007/BF01046094
author: Bernard Heinkel
published in: Journal of Theoretical Probability
publication date: 1990-01-01T00:00:00Z
review text: The author proves the following theorem: Let the random variable X take its values in a real, separable Banach space \((B,\| \cdot \|)\), equipped with its Borel \(\sigma\)-field, and \(\{X_ k\}\) be a sequence of independent copies of X. Then the statements (i) for all \(\alpha\in (0,1)\) the sequence \(V_ n=(1/A_ n^{\alpha})\sum^{n}_{k=0}A_{n-k}^{\alpha -1}X_ k\) converges a.s. to E(X), where \[ A_ 0^{\beta}=1,\quad A_ 1^{\beta}=\beta +1,...,\quad A_ k^{\beta}=(\beta +1)...(\beta +k)/k!,...\text{ for } every\quad real\quad \beta >-1. \] (ii) \(\| X\|^{1/\alpha}\) is integrable. are equivalent.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 4161761
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: Cesaro summability
zbMATH Keywords: exponential inequalities
Wikidata QID: Q112879217
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4161761