Bounds for the accuracy of Poissonian approximations of stable laws
type: Item
instance of: scholarly article
title: Bounds for the accuracy of Poissonian approximations of stable laws
zbMATH Open document ID: 0885.62014
DOI: 10.1016/S0304-4149(96)00101-9
published in: Stochastic Processes and their Applications
publication date: 1998-03-29T00:00:00Z
review text: Stable laws \(G_{\alpha }\) admit a well-known series representation of the type \[ G_{\alpha } = {\mathcal L} \;\Big (\sum ^{\infty }_{j=1} \Gamma _1^{-1/\alpha } X_j\Big ), \quad 0 < \alpha <2, \] where \(\Gamma _1, \Gamma _2,\ldots \) are the successive times of jumps of a standard Poisson process and \(X_1,X_2,\ldots \), denote i.i.d.\ random variables, independent of \(\Gamma _1,\Gamma _2,\ldots .\) We investigate the rate of approximation of \(G_{\alpha }\) by distributions of partial sums \(S_n = \sum ^n_{j=1} \Gamma ^{-1/\alpha }_j X_j\), and we obtain (asymptotically) optimal bounds for the variation of \(G_{\alpha } - {\mathcal L} \;(S_n)\). These results complement and improve results of A. Janicki and P. Kokoszka, and M. Ledoux and V. Paulauskas. Bounds for the concentration function of \(S_n\) are also proved.
reviewed by: Petr Lachout
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 1134788
zbMATH Keywords: stable laws
zbMATH Keywords: Poissonian representation
zbMATH Keywords: convergence in variation
zbMATH Keywords: convergence rates
zbMATH Keywords: Berry-Esseen bounds
zbMATH Keywords: concentration function
author: Vidmantas Bentkus
author: Friedrich Götze
author: V. Paulauskas
MaRDI profile type: Publication
cites work: On the lattice point problem for ellipsoids
cites work: Optimal rates of convergence in the CLT for quadratic forms
cites work: Q5563135
cites work: A Representation of Independent Increment Processes without Gaussian Components
cites work: Q5624460
cites work: Computer investigation of the Rate of Convergence of Lepage Type Series to α-Stable Random Variables
cites work: Q4869554
cites work: A rate of convergence in the Poissonian representation of stable distributions
cites work: Q3997990
cites work: Q3923314
cites work: Q4869556
cites work: Resampling permutations in regression without second moments
cites work: Strong and conditional invariance principles for samples attracted to stable laws
cites work: Convergence to a stable distribution via order statistics
cites work: Random Fourier Series with Applications to Harmonic Analysis. (AM-101)
cites work: Q5541664
cites work: Q5184198
cites work: On series representations of infinitely divisible random vectors
cites work: Q4301585
description: scientific article; zbMATH DE number 1134788