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Generalized Lorenz curves and convexifications of stochastic processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Youri Davydov  and
Ričardas Zitikis
Youri Davydov*
Affiliation:
Université des Sciences et Technologies de Lille
Ričardas Zitikis*
Affiliation:
University of Western Ontario
*
∗Postal address: Université des Sciences et Technologies de Lille, Laboratoire de Statistique et Probabilités, 59655 Villeneuve d'Ascq Cedex, France.
∗∗Postal address: University of Western Ontario, Department of Statistical and Actuarial Sciences, London, Ontario N6A 5B7, Canada. Email address: zitikis@stats.uwo.ca
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Abstract

We investigate convex rearrangements, called convexifications for brevity, of stochastic processes over fixed time intervals and develop the corresponding asymptotic theory when the time intervals indefinitely expand. In particular, we obtain strong and weak limit theorems for these convexifications when the processes are Gaussian with stationary increments and then illustrate the results using fractional Brownian motion. As a theoretical basis for these investigations, we extend some known, and also obtain new, results concerning the large sample asymptotic theory for the empirical generalized Lorenz curves and the Vervaat process when observations are stationary and either short-range or long-range dependent.

Keywords

Convex rearrangementsLorenz processVervaat processempirical processquantile processfractional Brownian motion

MSC classification

Primary: 60F05: Central limit and other weak theorems 60G17: Sample path properties
Secondary: 60G15: Gaussian processes 60G18: Self-similar processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 906 - 925
Copyright
Copyright © Applied Probability Trust 2003 

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