work [@Rosenblatt-1] for a study of quadratic variation of the Rosenblatt process.

Summarizing the asymptotic theory in the Gaussian case, we can conclude that the limiting behaviour in the framework of stationary increments Lévy driven moving averages is quite different. Not surprisingly, the quite stunning results of Theorem \[maintheorem\](i) and Theorem \[sec-order\](i) do not appear in the Gaussian setting (the convergence of the type may very well appear for differentiable Gaussian processes).

Limit theorems for discrete moving averages {#sec2.3}
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Asymptotic theory for statistics of discrete moving averages has been a subject of a deep investigation during the last thirty years. Indeed, the variety of different limiting distributions, which may appear under certain conditions on the innovations and weight coefficients, is quite astonishing. In a functional framework they include Brownian motion, $m$th order Hermite processes, stable Lévy processes with various stability indexes and fractional Brownian motion. We refer to the work [@at87; @hh97; @h99; @ks01; @s02; @s04] among many others. The limit theory is much more diverse and still not completely understood in contrast to the Gaussian case discussed in the previous subsection. For this reason, we will rather concentrate on some asymptotic results related to our set of conditions.

