On the asymptotic independence of the sum and rare values of weakly dependent stationary random variables
type: Item
instance of: scholarly article
title: On the asymptotic independence of the sum and rare values of weakly dependent stationary random variables
zbMATH Open document ID: 0848.60026
DOI: 10.1016/0304-4149(95)00054-2
author: Tailen Hsing
published in: Stochastic Processes and their Applications
publication date: 1996-10-08T00:00:00Z
review text: The author investigates the joint behavior of the sum and extreme order statistics of random variables in a large sample from many different angles. In the infinite variance case (in particular in the heavy tailed case) the extreme order statistics of large sample \(X_1, \dots, X_n\) dominate \(S_n\), therefore it is possible to describe the asymptotic distribution \(S_n\) in terms of those of the extreme order statistics, however it is possible for the lower and upper extremes to cancel one another in the limit so that the sum is asymptotically independent of the extremes. The author shows that if the stationary sequence \(\{X_i\}\) has finite variance and satisfies strong mixing condition, then the asymptotic distribution of \(S_n\) is unaffected by the information of whether the summands are in certain ``rare'' sets. An application of the result shows that \(S_n\) and the extremes are asymptotically independent. This is in sharp contrast to the infinite variance case.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 859947
zbMATH Keywords: central limit theorems
zbMATH Keywords: extreme value
zbMATH Keywords: strong mixing
zbMATH Keywords: point process
reviewed by: Adhir K. Basu
MaRDI profile type: Publication
full work available at URL: 0304-4149(95)00054-2
OpenAlex ID: W1967258484
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description: scientific article; zbMATH DE number 859947