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1 results

  • The nature of discrete second-order self-similarity

  • Part of
  • A. Gefferth, D. Veitch, I. Maricza, S. Molnár, I. Ruzsa
  • Journal:
    Advances in Applied Probability / Volume 35 / Issue 2 / June 2003
    Published online by Cambridge University Press:
    22 February 2016, pp. 395-416
    Print publication:
    June 2003

  • 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.