results on the power variations of such processes are presently available, exceptions to this are [@BCI Theorem 5.1] and [@sg14 Theorem 2]; see Remark \[rem3\] for a closer discussion of a result from [@BCI Theorem 5.1]. These two results are concerned with certain power variation of fractional Lévy process and have some overlad with our Theorem \[maintheorem\](ii) for the linear fractional stable motion, but we apply different proofs. The aim of this paper is to derive a rather complete picture of the first order asymptotic theory for power variation of the process $X$, and, in some cases, the associated second order limit theory. We will see that the type of convergence and the limiting random variables/distributions are quite surprising and novel in the literature. Apart from pure probabilistic interest, limit theory for power variations of stationary increments Lévy driven moving averages give rise to a variety of statistical methods (e.g.  identification and estimation of the unknown parameters $\alpha$ and $\beta$) and it provides a first step towards asymptotic theory for power variation of stochastic processes, which contain $X$ as a building block. In this context let us mention stochastic integrals with respect to $X$ and Lévy semi-stationary processes, which have 