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

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  • Mohamed Omari

    (Faculty of Sciences and Techniques
    Chouaib Doukkali University)

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.

Suggested Citation

  • Mohamed Omari, 2023. "An α-Order Fractional Brownian Motion with Hurst Index H ∈ (0,1) and α ∈ R + $\alpha \in \mathbbm {R}_{+}$," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 572-599, February.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:1:d:10.1007_s13171-021-00266-z
    DOI: 10.1007/s13171-021-00266-z
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    References listed on IDEAS

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    1. Azmoodeh, Ehsan & Sottinen, Tommi & Viitasaari, Lauri & Yazigi, Adil, 2014. "Necessary and sufficient conditions for Hölder continuity of Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 230-235.
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