of $V(B^H,p;k)_n$ is then $$\sum_{j=1}^\infty \rho_k^2 (j)<\infty,$$ where the power $2$ indicates the Hermite rank of $h$. The latter is obviously fulfilled for any $k\geq 2$ and also for $k=1$ if $H\in (0,3/4)$. The next result is a famous statement from [@BM83; @T79]. The Gaussian limit case is usually referred to as Breuer–Major central limit theorem.

\[mb-th\] The following assertions hold:

-   Assume that $k\geq 2$ or $k=1$ and $H\in (0,3/4)$. Then the central limit theorem holds $$\begin{aligned}
    \sqrt{n} \left( n^{-1+pH}V(B^H,p;k)_n - m_p \right) {\stackrel{d}{\longrightarrow}}\mathcal N(0, v_p), \end{aligned}$$ where $v_p= \sum_{l=2}^\infty l! \lambda_l^2 \Big(1+ 2 \sum_{j=1}^\infty \rho_k^l(j) \Big)$.

-   When $k=1$ and $H=3/4$ we have $$\begin{aligned}
    \frac{\sqrt{n}}{\log n} \left( n^{-1+pH}V(B^H,p;k)_n - m_p \right) {\stackrel{d}{\longrightarrow}}\mathcal N(0, \tilde{v}_p), \end{aligned}$$ where $\tilde{v}_p = 4 \lambda_2 \lim_{n\rightarrow \infty} \frac{1}{\log n} \sum_{j=1}^{n-1} \frac{n-k}{n} \rho_1^2(j)$.

-   When $k=1$ and $H\in (3/4,1)$ it holds that $$\begin{aligned}
    n^{2-2H} \left( n^{-1+pH}V(B^H,p;k)_n - m_p \right) {\stackrel{d}{\longrightarrow}}Z, \end{aligned}$$ where $Z$ is a Rosenblatt random variable.

The results of Theorem \[mb-th\] has been extended to the case of general stationary increments Gaussian processes in [@gl89]. Asymptotic theory for power variation of stochastic integrals with respect to Gaussian processes has been intensively studied in [@BNCP09; @BNCPW09; @cnw06]. We also refer to the interesting 