Limit laws for \(K\)-record times
type: Item
instance of: scholarly article
title: Limit laws for \(K\)-record times
zbMATH Open document ID: 0802.60027
DOI: 10.1016/0378-3758(94)90012-4
author: P. Deheuvels
author: V. B. Nevzorov
published in: Journal of Statistical Planning and Inference
publication date: 1994-12-15T00:00:00Z
review text: Let \(X_ 1, X_ 2, \dots, X_ n\) be independent and identically distributed random variables. Let \(X_{r,n}\) be the \(r\)-th order statistic of the \(X_ j\), \(1\leq j\leq n\). Assuming that the variables \(X_ j\) come from an infinite sequence, let the rank \(R_ n\) of \(X_ n\) be defined by \(X_ n= X_{R_ n, n}\). For a given sequence \(k_ n\), \(n\geq 1\), of positive integers, \(K= \{k_ n\}\), record times \(L(n)\), \(n\geq 0\), are defined by the sequential procedure \(L(0)=0\) and \(L(n)= \inf \{m\): \(m>L(n-1)\), \(R_ m\geq m-k_ m+1\}\) for \(n\geq 1\). The authors introduce indicator variables associated with the sequence \(L(n)\) which are shown to be independent. This enables the authors to establish strong laws of large numbers, functional laws of the iterated logarithm and strong invariance principles for \(L(n)\). These give far reaching extensions of the by now classical results for record times which correspond to the case \(k_ n=1\) for all \(n\geq 1\). Besides the reduction to independent variables, the authors utilize the martingale properties of several transforms of the sequence \(L(n)\) for evaluating some moments.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 605292
zbMATH Keywords: record times
zbMATH Keywords: order statistic
zbMATH Keywords: indicator variables
zbMATH Keywords: strong laws of large numbers
zbMATH Keywords: functional laws of the iterated logarithm
zbMATH Keywords: strong invariance principles
zbMATH Keywords: independent variables
zbMATH Keywords: martingale properties
reviewed by: Janos Galambos
MaRDI profile type: Publication
full work available at URL: 0378-3758(94)90012-4
OpenAlex ID: W2090008796
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description: scientific article; zbMATH DE number 605292