Asymptotic ordering of distribution functions and convolution semigroups
type: Item
instance of: scholarly article
title: Asymptotic ordering of distribution functions and convolution semigroups
zbMATH Open document ID: 0687.60018
DOI: 10.1007/BF02573252
author: Claudia Klüppelberg
published in: Semigroup Forum
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/135005
review text: Let \({\mathcal H}\) be the set of all distribution functions on [0,\(\infty)\), and define \[ {\mathcal L}(\gamma)=\{F\in {\mathcal H}:\quad \lim_{x\to \infty}(1-F(x-y))/(1-F(x))=e^{\gamma y},\quad y\in [0,\infty)\}, \] for \(\gamma\geq 0\). For \(F,G\in {\mathcal H}\), put \(F\leq^ wG\) whenever there is \(c\geq 1\) such that \(1-G(x)\leq c(1-F(x)),\) \(x\in [0,\infty).\) This paper contains a lot of easy results, such as: (a) if \(F_ 1\leq^ wF_ 2\) and \(G_ 1\leq^ wG_ 2\), then \(F_ 1*G_ 1\leq^ wF_ 2*G_ 2;\) (b) if \(F_ 1\leq^ wF_ 2\) and \(G_ 1\leq^ wG_ 2\), then \(pF_ 1+(1-p)G_ 1\leq^ wpF_ 2+(1-p)G_ 2\) for all \(p\in [0,1]\); (c) if \(F\in {\mathcal L}(\gamma)\) and \(G\in {\mathcal L}(\gamma ')\) for \(0\leq \gamma \leq \gamma '\), then \(pF+(1-p)G\in {\mathcal L}(\gamma)\) for all \(p\in (0,1]\).
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 4126388
zbMATH Keywords: convolution semigroups
zbMATH Keywords: convolution products
zbMATH Keywords: tail equivalence
reviewed by: Aurel Spătaru
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4126388