Let B = { B H ( t )} t ≥0 be a d -dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). Consider the functionals of k independent d -dimensional fractional Brownian motions $$\frac{1}{\sqrt{n}}\int_{0}^{e^{{nt}_1}} \cdot\cdot\cdot\int_{0}^{e^{{nt}_k}} f(B^{H,1}(s_1)+\cdot\cdot\cdot+B^{H,k}(s_k)){\rm{d}}s_1\cdot\cdot\cdot{{\rm{d}}s_k},$$ 1 n ∫ 0 e n t 1 · · · ∫ 0 e n t k f ( B H , 1 ( s 1 ) + · · · + B H , k ( s k ) ) d s 1 · · · 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 [19] of the case k = 1. It can also be regarded as a fractional generalization of Biane [3] in the case of Brownian motion.

中文翻译：

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多重独立分数布朗运动泛函的极限律

令 B = { BH ( t )} t ≥0 是具有 Hurst 参数 H ∈ (0, 1) 的 ad 维分数布朗运动。考虑 k 个独立的 d 维分数布朗运动的泛函 $$\frac{1}{\sqrt{n}}\int_{0}^{e^{{nt}_1}} \cdot\cdot\cdot\int_ {0}^{e^{{nt}_k}} f(B^{H,1}(s_1)+\cdot\cdot\cdot+B^{H,k}(s_k)){\rm{d }}s_1\cdot\cdot\cdot{{\rm{d}}s_k},$$ 1 n ∫ 0 ent 1 · · · ∫ 0 entkf ( BH , 1 ( s 1 ) + · · · + BH , k (sk)) ds 1 · · · dsk，其中 Hurst 指数 H = k/d。利用矩量法，证明了Xu[19]在k=1情况下的极限定律和扩展结果。也可以看作是Biane[3]在布朗运动情况下的分数推广。