On highly skewed fractional log‐stable noise sequences and their application.

Considering log‐LFSN (log‐linear fractional stable noise) sequences {Yn=eδ·Xn+ε}n∈ℤ, driven by non‐Gaussian one‐sided LFSN {Xn}n∈ℤ with constant skewness intensity β0∈[−1,1], for any δ∈ℝ−{0} and ε∈ℝ, we show that the auto‐covariance function (ACVF) {γY(h)}h∈ℤ exists if and only if {Xn}n∈ℤ is persistent, with stability index α∈(1,2), Hurst exponent H∈(1/α,1) and extreme skewness β0=−1 (if δ>0) or β0=1 (if δ<0). Within that range of existence, 1/2<1/α<H<1 and |β0|=1 in short, we calculate {γY(h)}h∈ℤ explicitly and establish persistence of {Yn}n∈ℤ too, by showing asymptotic proportionality of γY(h)≈|h|α·(H−1), as h→∞. We discuss explicit links of {γY(h)}h∈ℤ to a generalized co‐difference function of the driving one‐sided LFSN {Xn}n∈ℤ, and to the ACVF's of fractional Gaussian noise (FGN) and log‐FGN. The results are numerically demonstrated via ensemble simulation of synthetic time series generated by the considered log‐LFSN model fitted to time series of spatio‐temporal accumulations of rain rate data. [ABSTRACT FROM AUTHOR]