  In this paper we present some limit theorems for power variation of L\'evy
semi-stationary processes in the setting of infill asymptotics. L\'evy
semi-stationary processes, which are a one-dimensional analogue of ambit
fields, are moving average type processes with a multiplicative random
component, which is usually referred to as volatility or intermittency. From
the mathematical point of view this work extends the asymptotic theory
investigated in [14], where the authors derived the limit theory for $k$th
order increments of stationary increments L\'evy driven moving averages. The
asymptotic results turn out to heavily 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 $\alpha$. In this paper we will study the first order asymptotic
theory for L\'evy semi-stationary processes with a random
volatility/intermittency component and present some statistical applications of
the probabilistic results.
