  In this paper we study the large deviations of time averaged mean square
displacement (TAMSD) for Gaussian processes. The theory of large deviations is
related to the exponential decay of probabilities of large fluctuations in
random systems. From the mathematical point of view a given statistics
satisfies the large deviation principle, if the probability that it belongs to
a certain range decreases exponentially. The TAMSD is one of the main
statistics used in the problem of anomalous diffusion detection. Applying the
theory of generalized chi-squared distribution and sub-gamma random variables
we prove the upper bound for large deviations of TAMSD for Gaussian processes.
As a special case we consider fractional Brownian motion, one of the most
popular models of anomalous diffusion. Moreover, we derive the upper bound for
large deviations of the estimator for the anomalous diffusion exponent.
