Abstract: Let $B=\{B^H(t)\}_{t\geq0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$. Consider the functionals of $k$ independent $d$-dimensional fractional Brownian motions $$ \frac{1}{\sqrt{n}} \int^{e^{nt_1}}_0\cdots\int^{e^{nt_k}}_0 f(B^{H,1}(s_1)+\cdots +B^{H,k}(s_k)){\rm d}s_1\cdots{\rm d}s_k, $$ where the Hurst index $H=k/d$. Using the method of moments, we prove the limit law and extending a result by Xu \cite{xu} of the case $k=1$. It can also be regarded as a fractional generalization of Biane \cite{biane} in the case of Brownian motion.

Key words: Limit theorem, fractional Brownian motion, method of moments, chaining argument