An asymptotic evaluation of the tail of a multiple symmetric \(\alpha\)- stable integral
type: Item
instance of: scholarly article
title: An asymptotic evaluation of the tail of a multiple symmetric \(\alpha\)- stable integral
zbMATH Open document ID: 0689.60051
author: Gennady Samorodnitsky
author: Jerzy Szulga
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: Let Z be a symmetric Lévy \(\alpha\)-stable process on [0,1] with characteristic function \[ E \exp \{i\lambda Z(u)\}=\exp \{-u| \lambda |^{\alpha}\},\quad 0<\alpha <2, \] and let f be a real function on \([0,1]^ n\) vanishing on diagonals. Let \[ I_ n(f)=\int...\int f(t_ 1,...,t_ n)dZ(t_ 1)...dZ(t_ n) \] be a multiple stable integral (MSI). It is shown that under certain integrability conditions, \[ 2\lim_{x\to \infty}x^{\alpha}(\ln x)^{1-n}P(| I_ n(f)| >x)=\lim_{x\to \infty}x^{\alpha}(\ln x)^{1-n}P(I_ n(f)>x)= \] \[ =n\alpha^{n-1}(n!)^{\alpha -2}s^{- n}\int...\int | f(t_ 1,...,t_ n)|^{\alpha}dt_ 1...dt_ n \] where \(s=\int^{\infty}_{0}x^{-\alpha}\sin x x\). The proof is based on a LePage-type representation of MSI by multiple series of transformed arrival times of a Poisson process.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 4129754
zbMATH Keywords: stable Lévy process
zbMATH Keywords: multiple stable integral
zbMATH Keywords: LePage-type representation
zbMATH Keywords: Poisson process
reviewed by: Yuri Kabanov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991170
OpenAlex ID: W1993492072
DOI: 10.1214/AOP/1176991170
description: scientific article; zbMATH DE number 4129754