The nature of discrete second-order self-similarity

A. Gefferth, D. Veitch, I. Maricza, S. Molnár, I. Ruzsa

Research output: Contribution to journalArticle

5 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)395-416
Number of pages22
JournalAdvances in Applied Probability
Volume35
Issue number2
DOIs
Publication statusPublished - Jun 2003

Fingerprint

Self-similarity
Time series
Long-range Dependence
Fixed point
Regular Variation
Renormalization
Normalization
Power Law
Discrete-time
Fractional
Series
Alternatives
Operator

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability

Cite this

The nature of discrete second-order self-similarity. / Gefferth, A.; Veitch, D.; Maricza, I.; Molnár, S.; Ruzsa, I.

In: Advances in Applied Probability, Vol. 35, No. 2, 06.2003, p. 395-416.

Research output: Contribution to journalArticle

Gefferth, A, Veitch, D, Maricza, I, Molnár, S & Ruzsa, I 2003, 'The nature of discrete second-order self-similarity', Advances in Applied Probability, vol. 35, no. 2, pp. 395-416. https://doi.org/10.1239/aap/1051201654
Gefferth, A. ; Veitch, D. ; Maricza, I. ; Molnár, S. ; Ruzsa, I. / The nature of discrete second-order self-similarity. In: Advances in Applied Probability. 2003 ; Vol. 35, No. 2. pp. 395-416.
@article{c174e422ab04436ab90dba66f87eb649,
title = "The nature of discrete second-order self-similarity",
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.",
author = "A. Gefferth and D. Veitch and I. Maricza and S. Moln{\'a}r and I. Ruzsa",
year = "2003",
month = "6",
doi = "10.1239/aap/1051201654",
language = "English",
volume = "35",
pages = "395--416",
journal = "Advances in Applied Probability",
issn = "0001-8678",
publisher = "University of Sheffield",
number = "2",

}

TY - JOUR

T1 - The nature of discrete second-order self-similarity

AU - Gefferth, A.

AU - Veitch, D.

AU - Maricza, I.

AU - Molnár, S.

AU - Ruzsa, I.

PY - 2003/6

Y1 - 2003/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0038383051&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038383051&partnerID=8YFLogxK

U2 - 10.1239/aap/1051201654

DO - 10.1239/aap/1051201654

M3 - Article

AN - SCOPUS:0038383051

VL - 35

SP - 395

EP - 416

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 2

ER -