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The nature of discrete second-order self-similarity

Part of: Stochastic processes

Published online by Cambridge University Press:  22 February 2016

A. Gefferth ,
D. Veitch ,
I. Maricza ,
S. Molnár  and
I. Ruzsa
A. Gefferth*
Affiliation:
Budapest University of Technology and Economics
D. Veitch*
Affiliation:
University of Melbourne
I. Maricza*
Affiliation:
Budapest University of Technology and Economics
S. Molnár*
Affiliation:
Budapest University of Technology and Economics
I. Ruzsa*
Affiliation:
Alfréd Rényi Institute of Mathematics, Budapest
*
∗ Postal address: High Speed Networks Laboratory, Department of Telecommunications and Telematics, Budapest University of Technology and Economics, Magyar Tudósok körútja 2, H-1117 Budapest, Hungary.
∗∗ Postal address: Australian Research Council Special Research Center for Ultra-Broadband Information Networks, Department of Electrical and Electronic Engineering, University of Melbourne, VIC 3010, Australia. Email address: dveitch@unimelb.edu.au
∗ Postal address: High Speed Networks Laboratory, Department of Telecommunications and Telematics, Budapest University of Technology and Economics, Magyar Tudósok körútja 2, H-1117 Budapest, Hungary.
∗ Postal address: High Speed Networks Laboratory, Department of Telecommunications and Telematics, Budapest University of Technology and Economics, Magyar Tudósok körútja 2, H-1117 Budapest, Hungary.
∗∗∗ Postal address: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary.
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Abstract

A new treatment of second-order self-similarity and asymptotic self-similarity for stationary discrete time series is given, based on the fixed points of a renormalisation operator with normalisation factors which are not assumed to be power laws. A complete classification of fixed points is provided, consisting of the fractional noise and one other class. A convenient variance time function approach to process characterisation is used to exhibit large explicit families of processes asymptotic to particular fixed points. A natural, general definition of discrete long-range dependence is provided and contrasted with common alternatives. The closely related discrete form of regular variation is defined, its main properties given, and its connection to discrete self-similarity explained. Folkloric results on long-range dependence are proved or disproved rigorously.

Keywords

Discrete self-similaritylong-range dependencediscrete regular variation

MSC classification

Primary: 60G18: Self-similar processes
Secondary: 60G10: Stationary processes 60G12: General second-order processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 395 - 416
Copyright
Copyright © Applied Probability Trust 2003 

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