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

TL;DR

This paper studies non-ergodic stationary symmetric alpha-stable processes, deriving their spectral density estimation method using a fast periodogram-based approach that remains effective despite non-ergodicity.

Contribution

It introduces a spectral density estimator for non-ergodic processes using a novel periodogram method that is both consistent and computationally efficient.

Findings

Explicit non-ergodic limits of empirical characteristic functions derived

Spectral density estimator shown to be strongly consistent

Proposed method is fast, efficient, and unaffected by non-ergodicity

Abstract

We consider non-ergodic class of stationary real harmonizable symmetric $\alpha$-stable processes $X=\left\{X(t):t\in\mathbb{R}\right\}$ with a finite symmetric and absolutely continuous control measure. We refer to its density function as the spectral density of $X$. These processes admit a LePage series representation and are 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 which are i.i.d. with the (normalized) spectral density of $X$ as their probability density function. 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 $X$.