On highly skewed fractional log-stable noise sequences and their application

Abstract

Considering log-LFSN (log-linear fractional stable noise) sequences ${\{Y_n=e^{\delta ·X_n+\epsilon }\}}_{n\in \mathbb{Z}}$, driven by non-Gaussian one-sided LFSN ${\left\{X_n\right\}}_{n\in \mathbb{Z}}$ with constant skewness intensity ${\beta }_0\in [-1,1]$, for any $\delta \in ℝ-\left\{0\right\}$ and $\epsilon \in ℝ$, we show that the auto-covariance function (ACVF) ${\left\{{\gamma }_Y\right(h\left)\right\}}_{h\in \mathbb{Z}}$ exists if and only if ${\left\{X_n\right\}}_{n\in \mathbb{Z}}$ is persistent, with stability index $\alpha \in (1,2)$, Hurst exponent $H\in (1/\alpha ,1)$ and extreme skewness ${\beta }_0=-1$ (if $\delta >0$) or ${\beta }_0=1$ (if $\delta <0$). Within that range of existence, $1/2<1/\alpha <H<1$ and $\left|{\beta }_0\right|=1$ in short, we calculate ${\left\{{\gamma }_Y\right(h\left)\right\}}_{h\in \mathbb{Z}}$ explicitly and establish persistence of ${\left\{Y_n\right\}}_{n\in \mathbb{Z}}$ too, by showing asymptotic proportionality of ${\gamma }_Y\left(h\right)\approx |h{|}^{\alpha ·(H-1)}$, as $h\to \infty$. We discuss explicit links of ${\left\{{\gamma }_Y\right(h\left)\right\}}_{h\in \mathbb{Z}}$ to a generalized co-difference function of the driving one-sided LFSN ${\left\{X_n\right\}}_{n\in \mathbb{Z}}$, 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.