Power variation for a class of stationary increments Lévy driven moving averages

Abstract

In this paper, we present some new limit theorems for power variation of $k$th order increments of stationary increments Lévy 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 novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments $k\geq1$, the considered power $p>0$, the Blumenthal–Getoor index $\beta\in[0,2)$ of the driving pure jump Lévy 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 a second-order limit theorem connected to the ergodic type result. When the driving Lévy process $L$ is a symmetric $\beta$-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a $(k-\alpha)\beta$-stable totally right skewed random variable.