  Fractional Brownian motion (FBM) is the only Gaussian self-similar process
with stationary increments. Its increment process, called fractional Gaussian
noise, is ergodic and exhibits a property of power-like decaying
autocorrelation function (ACF) which leads to the notion of long memory. These
properties have made FBM important in modelling real-world data recorded in
different experiments ranging from biology to telecommunication. These
experiments are often disturbed by a noise which source can be just the
instrument error. In this paper we propose a rigorous statistical test based on
the ACF for FBM with added white Gaussian noise. 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 and comparing with other tests existing in the literature.
We also note that the introduced test procedure can be applied to an arbitrary
Gaussian process.
