Spectral representation of some non stationary alpha-stable processes

TL;DR

This paper introduces a new covariation spectral representation for certain non-stationary symmetric alpha-stable processes, broadening the theoretical framework beyond independence assumptions and analyzing their non-stationarity characteristics.

Contribution

It generalizes the covariation spectral representation to processes with weaker conditions than independence, including non-stationary and harmonisable SαS processes.

Findings

Developed a covariation spectral representation under weaker conditions.

Analyzed non-stationarity in harmonisable SαS processes.

Characterized processes with periodic or almost-periodic covariation functions.

Abstract

In this paper, we give a new covariation spectral representation of some non stationary symmetric $\alpha$-stable processes (S$\alpha$S). This representation is based on a weaker covariation pseudo additivity condition which is more general than the condition of independence. This work can be seen as a generalization of the covariation spectral representation of processes expressed as stochastic integrals with respect to independent increments S$\alpha$S processes (see Cambanis (1983)) or with respect to the general concept of independently scattered S$\alpha$S measures (Samorodnitsky and Taqqu 1994). Relying on this result we investigate the non stationarity structure of some harmonisable S$\alpha$S processes especially those having periodic or almost-periodic covariation functions.