In this paper we establish limit theorems for power variations of stochastic processes controlled by fractional Brownian motions with Hurst parameter $H\leq 1/2$. We show that the power variations of such processes can be decomposed into the mix of several weighted random sums plus some remainder terms, and the convergences of power variations are dominated by different combinations of those weighted sums depending on whether $H<1/4$, $H=1/4$, or $H>1/4$. We show that when $H\geq 1/4$ the centered power variation converges stably at the rate $n^{-1/2}$, and when $H<1/4$ it converges in probability at the rate $n^{-2H}$. We determine the limit of the mixed weighted sum based on a rough path approach developed in \cite{LT20}.

翻译：本文建立了由Hurst参数$H\leq 1/2$的分数布朗运动驱动的随机过程的幂变差极限定理。我们证明此类过程的幂变差可分解为若干加权随机和与余项的混合，且幂变差的收敛性由这些加权和的不同组合主导，具体取决于$H<1/4$、$H=1/4$或$H>1/4$的情形。研究表明，当$H\geq 1/4$时，中心化幂变差以$n^{-1/2}$的速率稳定收敛；当$H<1/4$时，则以$n^{-2H}$的速率依概率收敛。基于文献\cite{LT20}发展的粗糙路径方法，我们确定了混合加权和的极限形式。