Theory of Probability &Its Applications, Volume 65, Issue 1, Page 32-48, January 2020.
Sample statistics of samples from a fractional Brownian motion with Hurst exponent $H$, and in particular, autocovariance statistics, are considered. Two statistics characterizing the covariate dependence between the increments of this process are studied; in particular, their asymptotic properties and the limit distributions are examined. Each of the statistics is shown to converge almost everywhere; their limits are evaluated. It is found that these statistics have different limit distributions depending on the value of $H$. A complete description of these distributions in terms of semi-invariants is put forward.

中文翻译：

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分数布朗运动函数的极限定理

概率论及其应用，第65卷，第1期，第32-48页，2020年1月
。考虑了具有Hurst指数$ H $的分数布朗运动样本的样本统计，尤其是自协方差统计。研究了表征该过程增量之间的协变量相关性的两个统计量；特别是检查了它们的渐近性质和极限分布。每个统计数据几乎都可以汇聚到一起。评估其极限。发现这些统计量具有不同的限制分布，具体取决于$ H $的值。提出了关于这些分布的半不变性的完整描述。