$0$. Recall that the process $\widetilde{X}$ is the linear fractional stable motion. In particular, under the assumptions of Theorem \[maintheorem\](ii), it is a linear fractional stable motion with $\beta$-stable marginals and self-similarity index $H=\alpha +1/\beta$. Thus, one may transfer the first order asymptotic theory for power variation of $\widetilde{X}$ to the corresponding results for power variation of $X$. However, the law of large numbers for power variation of $\widetilde{X}$ is easier to handle than the original statistic due to self-similarity property of $\widetilde X$, which allows to transform the original triangular observation scheme into a usual one when studying distributional properties. Then, the standard ergodic limit theory becomes applicable. Indeed, this is exactly the method of proof of Theorem \[maintheorem\](ii). We remark however that it is much more technical to use the relationship between $X$ and $\widetilde{X}$ for the proof of the second order asymptotic theory in Theorem \[sec-order\]. In fact, we use a more direct approach to show the results of Theorem \[sec-order\].

Limit theory in the Gaussian case {#sec2.2}
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Throughout this subsection we recall the asymptotic theory for power variation of fractional Brownian motion $(B_t^H)_{t\geq 0}$ with Hurst parameter $H\in (0,1)$ and relate it to our limit 