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

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