Fractional Brownian motion (FBM) is related to the notions of self-similarity, ergodicity and long memory. These properties have made FBM important in modeling real-world phenomena in different experiments ranging from telecommunication to biology. However, these experiments are often disturbed by a noise which source can be, e.g., the instrument error. In this paper we propose a rigorous statistical test for FBM with added white Gaussian noise which is based on the autocovariance function. To this end we derive a distribution of the test statistic which is given explicitly by the generalized chi-squared distribution. This allows us to find critical regions for the test with a given significance level. We check the quality of the introduced test by studying its power for alternatives being FBM’s with different self-similarity parameters and the scaled Brownian motion which is also Gaussian and self-similar. We note that the introduced test can be adapted to an arbitrary Gaussian process with a given covariance structure.

分数布朗运动（FBM）与自相似性，遍历性和长记忆性有关。这些特性使FBM在模拟从电信到生物学的不同实验中对现实现象进行建模时很重要。但是，这些实验通常会受到噪声的干扰，该噪声源可能是仪器误差。在本文中，我们提出了基于自协方差函数的，添加了白高斯噪声的FBM的严格统计检验。为此，我们得出了检验统计量的分布，该分布由广义卡方分布明确给出。这使我们能够找到具有给定显着性水平的测试关键区域。我们通过研究引入的测试的能力来检验所引入测试的质量，这些测试是具有不同自相似性参数的FBM以及缩放的布朗运动（也是高斯和自相似）的替代方法。我们注意到引入的检验可以适应给定协方差结构的任意高斯过程。