On limit theory for functionals of stationary increments Levy driven moving averages

Abstract

In this paper we obtain new limit theorems for variational functionals of high frequency observations of stationary increments L\'evy driven moving averages. We will see that the asymptotic behaviour of such functionals heavily depends on the kernel, the driving L\'evy process and the properties of the functional under consideration. We show the "law of large numbers" for our class of statistics, which consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we also prove the associated weak limit theory, which again consists of three different cases. Our work is related to [9,10], who considered power variation functionals of stationary increments L\'evy driven moving averages.