Limit theorems for a class of stationary increments Levy driven moving averages

TL;DR

This paper establishes new limit theorems for power variation of stationary increments Lévy-driven moving averages, revealing complex asymptotic behaviors depending on parameters like increment order, power, and Lévy process characteristics.

Contribution

It introduces novel limit theorems for power variation in Lévy-driven moving averages, highlighting the dependence on multiple parameters and the surprising asymptotic regimes.

Findings

First order limits include stable convergence, ergodic limits, and convergence in probability.

Second order limit theorem related to ergodic behavior.

Special case for symmetric β-stable Lévy processes yields CLT and stable limits.

Abstract

In this paper we present some new limit theorems for power variation of $k$th order increments of stationary increments L\'evy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives very surprising results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we will show that the first order limit theorems and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal--Getoor index $\beta \in (0,2)$ of the driving pure jump L\'evy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $\alpha $. First order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove the second order limit theorem connected to the ergodic type result. When the driving L\'evy process $L$ is a symmetric $\beta$-stable process we obtain two different limits: a central limit theorem and convergence in distribution towards a $(1-\alpha )\beta$-stable totally right skewed random variable.