An α-Order Fractional Brownian Motion with Hurst Index H ∈ (0,1) and α ∈ R + $\alpha \in \mathbbm {R}_{+}$

Mohamed Omari ()
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Mohamed Omari: Faculty of Sciences and Techniques

Sankhya A: The Indian Journal of Statistics, 2023, vol. 85, issue 1, No 22, 572-599

Abstract: Abstract This paper provides an α-order fractional Brownian motion (α-fBm) with Hurst index H ∈ (0,1) and (hereafter Z H α ( t ) , t ≥ 0 $Z_{H}^{\alpha } (t),~t\geq 0$ ), as extension of the n th fBm where n is a nonnegative integer and H ∈ (n − 1,n) (e.g. Perrin et al. IEEE Trans. Signal Process., 49, 1049–1059, 2001). We show that the process Z H α ( t ) $Z_{H}^{\alpha } (t)$ is (H + α)-self-similar and satisfies the long-range dependence property. The covariance function and single-trajectory power spectral density (PSD) are also examined. Finally, via an illustrative example we discuss the impact of the order α on procedures of estimation.

Keywords: n th order fractional Brownian motion; self-similarity; long-range dependence; power spectral density; parametric estimation.; Primary 60G18; Secondary 60G17 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13171-021-00266-z

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