An operator theory approach to the evanescent part of a two-parametric weak-stationary stochastic process

TL;DR

This paper introduces an operator theory method to analyze the evanescent component of a two-parameter weak-stationary stochastic process, emphasizing its dependence on the shape of the past, advancing understanding of process decomposition.

Contribution

It presents a novel operator theory approach to characterize the evanescent part of two-dimensional weak-stationary processes, focusing on its dependence on past shape.

Findings

The evanescent part is uniquely characterized using operator theory.

The shape of the past influences the evanescent component.

Provides a new perspective on process decomposition in stochastic analysis.

Abstract

A new approach to the evanescent part of a two-dimensional weak-stationary stochastic process with the past given by a half-plane is proceed. The classical result due to Helson and Lowdenslager divides a two-parametric weak-stationary stochastic process into three parts. In this paper we describe the most untouchable one - the evanescent part. Moreover, we point out how this part depends on the shape of the past.