Non-ergodic inference for stationary-increment harmonizable stable processes

TL;DR

This paper studies stationary-increment harmonizable stable processes with infinite control measure, providing conditions for path integrability, distributional characteristics, and constructing consistent estimators for key parameters like stability index and Hurst parameter.

Contribution

It introduces new methods for analyzing these processes, including path integrability conditions and estimators derived from periodogram estimates.

Findings

Conditions for path integrability with respect to finite measures

Distributional characteristics of path integrals

Consistent estimators for stability index and Hurst parameter

Abstract

We consider the class of stationary-increment harmonizable stable processes with infinite control measure, which most notably includes real harmonizable fractional stable motions. We give conditions for the integrability of the paths of such processes with respect to a finite, absolutely continuous measure and derive the distributional characteristics of the path integral with respect to said measure. The convolution of the path of a stationary-increment harmonizable stable process with a suitable measure yields a real stationary harmonizable stable process with finite control measure. This allows us to construct consistent estimators for the index of stability as well as the kernel function in the integral representation of a stationary increment harmonizable stable process (up to a constant factor). For real harmonizable fractional stable motions consistent estimators for the index of stability and its Hurst parameter are given. These are computed directly from the periodogram frequency estimates of the smoothed process.