Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric $α$-stable processes

We consider stationary real harmonizable symmetric $\alpha$-stable processes $X=\left\{X(t):t\in\mathbb{R}\right\}$ with a finite control measure. Assuming the control measure is symmetric and absolutely continuous with respect to the Lebesgue measure on the real line, we refer to its density function as the spectral density of $X$. Standard methods for statistical inference on stable processes cannot be applied as harmonizable stable processes are non-ergodic. A stationary real harmonizable symmetric $\alpha$-stable process $X$ admits a LePage series representation and is conditionally Gaussian which allows us to derive the non-ergodic limit of sample functions on $X$. In particular, we give an explicit expression for the non-ergodic limits of the empirical characteristic function of $X$ and the lag process $\left\{X(t+h)-X(t):t\in\mathbb{R}\right\}$ with $h>0$, respectively. The process admits an equivalent representation as a series of sinusoidal waves with random frequencies whose probability density function is in fact the (normalized) spectral density of $X$. Based on strongly consistent frequency estimation using the periodogram we present a strongly consistent estimator of the spectral density. The periodogram's computation is fast and efficient, and our method is not affected by the non-ergodicity of $X$. Most notably no prior knowledge on parameters of the process such as its index of stability $\alpha$ is needed.