Large deviations of inverse processes with nonlinear scalings
type: Item
instance of: scholarly article
title: Large deviations of inverse processes with nonlinear scalings
zbMATH Open document ID: 0939.60011
author: N. G. Duffield
author: Ward Whitt
published in: The Annals of Applied Probability
publication date: 1999-11-29T00:00:00Z
review text: For a right-continuous, nonnegative, nondecreasing stochastic process \((Z_t)_{t\geq 0}\) with \(\lim_{t\to\infty }Z_t=\infty \) we define the inverse process \(T_z=\inf\{t\geq 0, Z_t\geq z\}\), \(z\geq 0\), which is a left-continuous, nonnegative, nondecreasing process with \(T_0=0\) and \(\lim_{t\to\infty }T_t=\infty \). Then, under regularity conditions, \(u(Z_t)/w(t)\) satisfies the large deviation principle (LDP) with a scaling function \(v(t)\) and a rate function \(I\) if and only if \(w(T_z)/u(z)\) satisfies the LDP with the scaling function \(v\circ w^{-1}\circ u(z)\) and the rate function \(J(x)=f(x)I(1/x)\) where \(f(x)=x^{i(f)}\) with \(i(f)\) being the index of the regularly varying function \(v\circ w^{-1}\). An application to queueing models as well as examples concerning fractional Gaussian noise, time-transformed stationary processes, compound processes and renewal theory are also given.
reviewed by: Ilya Pavlyukevich
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60G18
zbMATH DE Number: 1319904
zbMATH Keywords: large deviations
zbMATH Keywords: renewal theory
zbMATH Keywords: queueing theory
zbMATH Keywords: regularly varying functions
zbMATH Keywords: inverse processes
zbMATH Keywords: counting processes
zbMATH Keywords: fractional Gaussian noise
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1028903372
OpenAlex ID: W1990016613
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DOI: 10.1214/AOAP/1028903372
description: scientific article; zbMATH DE number 1319904