A limit theorem for a class of stationary increments L\'{e}vy moving average process with multiple singularities

TL;DR

This paper establishes new limit theorems for power variations of stationary increment Lévy moving average processes with multiple singularities in the kernel function, extending previous results that focused on a single singularity.

Contribution

It extends existing asymptotic results to cases where the kernel function has multiple singularities, broadening the understanding of Lévy driven processes.

Findings

Derived limit theorems for processes with multiple singularities

Showed how multiple singularities affect asymptotic behavior

Connected results to Brownian semi-stationary models

Abstract

In this paper we present some new limit theorems for power variations of stationary increment L\'{e}vy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477--4528, Festschrift for Bernt {\O}ksendal, Stochastics 81(1) (2017), 360--383] under the assumption that the kernel function potentially exhibits a singular behaviour at $0$. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article [Stoch. Process. Appl. 125(2) (2014), 653--677] that studied the same mathematical question for the class of Brownian semi-stationary models.