The rate of convergence for subexponential distributions
type: Item
instance of: scholarly article
title: The rate of convergence for subexponential distributions
zbMATH Open document ID: 0935.60005
DOI: 10.1007/BF02465540
author: K. Appert
published in: Lithuanian Mathematical Journal
publication date: 2000-05-04T00:00:00Z
review text: Let \(F\) be a subexponential distribution function (i.e. \((1-F^{*n} (x))/(1-F(x)) \to n\) as \(x\to\infty\) for all \(n\geq 2)\), assume that \(F\) has a density \(f\), and define the hazard rate function \(q(x)=f(x)/r(x)\), where \(r(x)=1 -F(x)\), \(x>0\). Further, set \(R(x)=\int^x_0r(y)dy\) and \(U(x)=r^2 (x/2)/f (x)R(x)\), \(x>0\). A typical result of this paper, dealing with the asymptotic behaviour of the difference \(R_n(x)= 1-F^{*n}(x)- n(1-F(x))\), is as follows: Assume that \(\lim\sup_{x\to\infty} q(xy)/q(x) <\infty\) for any \(y>0\), that \(\int^x_0 r(x-y)r(y)dy =O(1)r(x) R(x)\) as \(x\to\infty\), and that \(U(x)=O(1)\) as \(x\to\infty\). Then, for \(n\geq 2\), \(R_n(x)=O(1)f(x)R(x)\) as \(x\to\infty\). The authors discuss also applications of their results to the distance between the distribution functions of sums and maxima of i.i.d. random variables, to transient renewal theory, and to the relation between the tail of an infinitely divisible distribution function and its Lévy measure.
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 1262929
zbMATH Keywords: transient renewal theory
zbMATH Keywords: infinitely divisible distribution function
zbMATH Keywords: Lévy measure
reviewed by: Aurel Spătaru
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1262929