Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications
type: Item
instance of: scholarly article
title: Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications
zbMATH Open document ID: 1184.62099
author: Jun An
author: De-Mei Yuan
published in: Science in China. Series A
publication date: 2009-12-11T00:00:00Z
review text: A sequence of random variables \(\{X_i, 1 \leq i \leq n \}\) is called negatively asociated (NA) if for every pair of disjoint subsets \( A \) and \( B\) of \(\{ 1, 2, \dots , n \}\), \[ \text{Cov}(f(X_i, i \in A), g(X_j , j \in B)) \leq 0, \] whenever \(f\) and \(g\) are coordinatewise nondecreasing and the covariance exists. A sequence of random variables \(\{ X_n\), \(n\geq 1\}\) is called asymptotically almost negatively associated (AANA) if there exists a nonnegative sequence \(q(n) \rightarrow 0 \) as \( n \rightarrow \infty\) such that \[ \text{Cov}(f(X_n), g(X_{n+1}, \dots , X_{n+k} )) \leq q(n)(\text{Var}(f(X_n)) \text{Var}(g(X_{n+1}, \dots ,X_{n+k})))^{1/2}, \] for all \( n, k \geq 1 \) and for all coordinatewise nondecreasing continuous functions \(f\) and \(g\) whenever the variances exist. For NA random variables a lot of sharp and elegant estimates are available. Some Rosenthal type moment inequalities are also introduced. For AANA random variables, some excellent results are also available. However, for AANA random variables, Rosenthal type inequalities are not yet available. The authors establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As application of these inequalities, by employing the notions of residual Cesàro \(\alpha\)-integrability and strong residual Cesàro \(\alpha\)-integrability, they derive some results on \(L_p\)-convergence, where \(1 < p < 2\), and on complete convergence. In addition, they estimate the rate of convergence in the Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.
reviewed by: Tae Il Jeon
Mathematics Subject Classification ID: 62H20
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 62H05
zbMATH DE Number: 5645732
zbMATH Keywords: asymptotically almost negative association
zbMATH Keywords: residual Cesàro \(\alpha\)-integrability
zbMATH Keywords: strong residual Cesàro \(\alpha\)-integrability
zbMATH Keywords: \(L_{p}\)-convergence
zbMATH Keywords: complete convergence
zbMATH Keywords: Marcinkiewicz-Zygmand strong law
MaRDI profile type: Publication
full work available at URL: s11425-009-0154-z
OpenAlex ID: W1994100571
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DOI: 10.1007/S11425-009-0154-Z
description: scientific article; zbMATH DE number 5645732