Methodology and Computing in Applied Probability

, Volume 7, Issue 3, pp 379–400 | Cite as

Simulation of Weakly Self-Similar Stationary Increment \({\text{Sub}}_{\varphi } {\left( \Omega \right)}\)-Processes: A Series Expansion Approach

Article

Abstract

We consider simulation of \({\text{Sub}}_{\varphi } {\left( \Omega \right)}\)-processes that are weakly selfsimilar with stationary increments in the sense that they have the covariance function
$$R{\left( {t,s} \right)} = \frac{1}{2}{\left( {t^{{2H}} + s^{{2H}} - {\left| {t - s} \right|}^{{2H}} } \right)}$$
for some H ∈ (0, 1). This means that the second order structure of the processes is that of the fractional Brownian motion. Also, if \(H >\frac{1} {2}\) then the process is long-range dependent.

The simulation is based on a series expansion of the fractional Brownian motion due to Dzhaparidze and van Zanten. We prove an estimate of the accuracy of the simulation in the space C([0, 1]) of continuous functions equipped with the usual sup-norm. The result holds also for the fractional Brownian motion which may be considered as a special case of a \({\text{Sub}}_{{{x^{2} } \mathord{\left/ {\vphantom {{x^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( \Omega \right)}\)-process.

Keywords

fractional Brownian motion φ-sub-Gaussian processes long-range dependence self-similarity series expansions simulation 

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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Yuriy Kozachenko
    • 1
  • Tommi Sottinen
    • 2
  • Olga Vasylyk
    • 1
  1. 1.Mechanics and Mathematics Faculty, Department of Probability Theory and Math. StatisticsTaras Shevchenko Kyiv National UniversityKyivUkraine
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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