theory. The main demonstrated results have been established in the classical work [@BM83; @T79]. We write $V(B^H,p;k)_n$ to denote the power variation statistics defined at associated with the fractional Brownian motion $(B_t^H)_{t\geq 0}$.

First of all, we observe the law of large numbers $$\begin{aligned}
n^{-1+pH}V(B^H,p;k)_n {\stackrel{{\mathbb{P}}}{\longrightarrow}}m_p:= {\mathbb{E}}[|n^H\Delta_{i,k}^{n} B^H|^p], \end{aligned}$$ which follows from the ergodic theorem (note that $m_p$ is independent of $n$ due to self-similarity property of $B^H$). The associated weak limit theory depends on the interplay between the correlation kernel of the fractional Brownian noise and the *Hermite rank* of the function $h(x)=|x|^p - m_p$. Recall that the correlation kernel $\rho_k(j)=\text{corr}(n^H\Delta_{k,k}^{n} B^H, n^H\Delta_{k+j,k}^{n} B^H)$ of $k$th order differences of the fractional Brownian motion satisfies that $$|\rho_k(j)|\leq K j^{2H-2k} \qquad \text{for }j\geq 1,$$ for some $K>0$. The Hermite expansion of the function $h$ is defined as $$h(x)=|x|^p - m_p = \sum_{l=2}^{\infty} \lambda_l H_l(x),$$ where $(H_l)_{l\geq 0}$ are Hermite polynomials, i.e.  $$H_0(x)=1 \quad \text{and} \quad H_l (x)= (-1)^l \exp(x^2/2) \frac{d^l}{dx^l} \{-\exp(x^2/2)\} \quad \text{for } l\geq 1.$$ The Hermite rank of $h$ is the smallest index $l$ with $\lambda_l \not =0$, which is $2$ in our case. The condition for the validity of a central limit theorem associated to a standardized version 