  Most real-world systems exhibit a multiscale behaviour that needs to be taken
into consideration when fitting the effective dynamics to data sampled at a
given scale. In the case of stochastic multiscale systems driven by Brownian
motion, it has been shown that in order for the Maximum Likelihood Estimators
of the parameters of the limiting dynamics to be consistent, data needs to be
subsampled at an appropriate rate. Recent advances in extracting effective
dynamics for fractional multiscale systems make the same question relevant in
the fractional diffusion setting. We study the problem of parameter estimation
of the diffusion coefficient in this context. In particular, we consider the
multiscale fractional Ornstein-Uhlenbeck system (fractional kinetic Brownian
motion) and we provide convergence results for the Maximum Likelihood Estimator
of the diffusion coefficient of the limiting dynamics, using multiscale data.
To do so, we derive asymptotic bounds for the spectral norm of the inverse
covariance matrix of fractional Gaussian noise.
