A limit theorem for a class of stationary increments Lévy moving average process with multiple singularities

Abstract

In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477-4528, Festschrift for Bernt Øksendal, 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.

Local asymptotic self-similarity for heavy-tailed harmonizable fractional Lévy motions

“…The most well-known representative of the first class is the linear fractional stable motion (lfsm), whose probabilistic and statistical properties have been studied in numerous articles. We refer to [4,5,6,7,17,22,23] for a detailed exposition of limit theorems for functionals of lfsm, and to [1,10,12,18,19,21] for statistical estimation of lfsm and related processes.…”