It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter $d$ converges weakly to fractional Brownian motion for $d>1/2$. We show that derivatives of order $m=1,2,\ldots$ of the normalized fractional process with respect to the fractional parameter $d$, converge weakly to the corresponding derivatives of fractional Brownian motion. As an illustration we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.

中文翻译：

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对分数布朗运动导数的弱收敛

众所周知，在合适的规律性条件下，带有分数参数$d$ 的归一化分数过程在$d>1/2$ 时弱收敛到分数布朗运动。我们表明，归一化分数过程关于分数参数 $d$ 的阶数 $m=1,2,\ldots$ 弱收敛到分数布朗运动的相应导数。作为说明，我们将结果应用于多分数向量自回归模型中分数向量的渐近分布。