  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.
