A note on tightness
type: Item
instance of: scholarly article
title: A note on tightness
zbMATH Open document ID: 0859.60022
DOI: 10.1016/0167-7152(95)00094-1
author: Christian Genest
author: Kilani Ghoudi
author: Bruno Rémillard
published in: Statistics \& Probability Letters
publication date: 1997-04-09T00:00:00Z
review text: Let \(T=\{(s,t):0\leq s\leq t\leq 1\}\), and let \(\mathcal T\) be the set of functions from \(T\) to \([0,\infty)\) that are decreasing in the first argument and increasing in the second one. Two concepts: ``a function \(\varphi\in{\mathcal T}\) is said to be subservient to another function \(\rho\in{\mathcal T}\)'', and ``a function \(\rho\in{\mathcal T}\) is said to be diagonally vanishing'' are introduced (the details are in the paper). In terms of those concepts the Billingsley's classical tightness criterion for sequences of càdlàg processes on \([0,1]\) is extended to the following: \[ P[|X_n(u)-X_n(s)|\geq\lambda, |X_n(t)-X_n(u)|\geq\lambda]\leq\lambda^{-2\gamma}\varphi_n(s,t), \] \(\lambda>0\), \(n\geq 1\), \(0\leq s\leq u\leq t\leq 1\), \(t-s<\Delta\), for some \(\Delta>0\), \(\gamma>0\) and a sequence \((\varphi_n)\) of elements of \(\mathcal T\) such that \(\varphi_n\) is \((1/(2\gamma+1),\Delta)\)-subservient to \(\rho_n\) for all \(n\) superior to some \(n_0\), and that the sequence \((\rho_n)\) converges pointwise to a limiting diagonally vanishing function \(\rho\in{\mathcal T}\).
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G99
zbMATH DE Number: 896097
zbMATH Keywords: tightness criterion
zbMATH Keywords: càdlàg processes
reviewed by: Dominik Szynal
MaRDI profile type: Publication
full work available at URL: 0167-7152(95)00094-1
OpenAlex ID: W2129437183
cites work: Q5560061
cites work: On the central limit theorem in \(D[0,1]\)
cites work: Central limit theorems in D[0, 1]
cites work: Q4856610
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description: scientific article; zbMATH DE number 896097