On integral representations of operator fractional Brownian fields. (English) Zbl 1294.60064
Summary: Operator fractional Brownian fields (OFBFs) are Gaussian, stationary-increment vector random fields that satisfy the operator self-similarity relation \(\{X(c^E t)\}_{t \in \mathbb{R}^m} \overset{\mathcal{L}} {=} \{c^H X(t) \}_{t \in \mathbb{R}^m}\). We establish a general harmonizable representation (Fourier domain stochastic integral) for OFBFs. Under additional assumptions, we also show how the harmonizable representation can be re-expressed as a moving average stochastic integral, thus answering an open problem described in H. Biermé et al. [Stochastic Processes Appl. 117, No. 3, 312–332 (2007; Zbl 1111.60033)].
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60G18 | Self-similar stochastic processes |
| 60G15 | Gaussian processes |
Keywords:
operator fractional Brownian fields; operator scaling; operator self-similarity; harmonizable representations; moving average representations; anisotropyCitations:
Zbl 1111.60033References:
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