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Rezakhah S, Philippe A, Modarresi N. Innovative methods for modeling of scale invariant processes. COMMUN STAT-THEOR M 2018. [DOI: 10.1080/03610926.2017.1350273] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- S. Rezakhah
- Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
| | - A. Philippe
- Laboratoire de Mathématiques Jean Leray, 2 rue de la houssinire, Université de Nantes, Nantes Cedex 3, France
| | - N. Modarresi
- Faculty of Mathematics and computer science, Allameh Tabataba’i University, Tehran, Iran
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Davydov Y, Zitikis R. Generalized Lorenz curves and convexifications of stochastic processes. J Appl Probab 2016. [DOI: 10.1239/jap/1067436090] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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.
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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.
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