Some probabilistic inequalities for sums of random elements
type: Item
instance of: scholarly article
title: Some probabilistic inequalities for sums of random elements
zbMATH Open document ID: 0698.60020
DOI: 10.1007/BF01104814
author: T. V. Bychkova
published in: Journal of Soviet Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let \(X_ k\), \(k=1,2,...\), be independent random elements in a separable Hilbert space H. Assume that \(E X_ k=0\) and \(\| X_ k\| \leq C E \| X_ k\|\) a.s., \(C>0\), \(k=1,2,... \). The paper is devoted to the study of probability inequalities of the Bernstein-Prokhorov-type [cf. \textit{Yu. V. Prokhorov}, Theory Probab. Appl. 4, 201-203 (1960); translation from Teor. Veroyatn. Primen. 4, 211-214 (1959; Zbl 0093.151)] for the infinite sum \(X_ 1+X_ 2+...\), if the additional condition \[ \sum^{\infty}_{k=1}E \| X_ k\|^{\alpha}<\infty \] for some \(\alpha,\) \(1<\alpha \leq 2,\) is fulfilled. Also, a generalization to unbounded random elements is obtained: \[ P(\| \sum^{\infty}_{k=1}a_ kX_ k\| >x)\leq \exp (- C_{\alpha}x^{\alpha /(\alpha -1)}),\quad C_{\alpha}>0. \] (Here we omit the exact assumptions).
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 4145029
zbMATH Keywords: Hilbert space valued random elements
zbMATH Keywords: sums of independent random variables
zbMATH Keywords: probability inequalities of the Bernstein-Prokhorov-type
zbMATH Keywords: unbounded random elements
reviewed by: Ludwig Paditz
MaRDI profile type: Publication
cites work: Certain properties of random series
cites work: Q4195701
cites work: DIRICHLET SERIES WITH INDEPENDENT EXPONENTS AND SOME OF THEIR APPLICATIONS
cites work: Q3254471
description: scientific article; zbMATH DE number 4145029