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

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.