## Simulation of weakly self-similar stationary increment $$\mathbf{Sub}_\varphi(\Omega)$$-processes: A series expansion approach.(English)Zbl 1082.60512

Summary: We consider simulation of $$\text{Sub}_\varphi(\Omega)$$-processes that are weakly selfsimilar with stationary increments in the sense that they have the covariance function $R(t,s) = \frac12 \left(t^{2H} + s^{2H} - | t-s|^{2H} \right)$ for some $$H \in (0,1)$$. This means that the second order structure of the processes is that of the fractional Brownian motion. Also, if $$H > \frac12$$, then the process is long-range dependent. The simulation is based on a series expansion of the fractional Brownian motion due to K. Dzhaparidze and H. van Zanten [Probab. Theory Relat. Fields 130, No. 1, 39–55 (2004; Zbl 1059.60048)]. 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 $$\text{Sub}_{x^2/2}(\Omega)$$-process.

### MSC:

 60G18 Self-similar stochastic processes 60G15 Gaussian processes 68U20 Simulation (MSC2010) 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$

Zbl 1059.60048

longmemo
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