Probability distribution of the time-averaged mean-square displacement of a Gaussian process

We study the probability distribution of the time-averaged mean-square displacement of a discrete Gaussian process. An empirical approximation for the probability density is suggested and numerically validated for fractional Brownian motion. The optimality of quadratic forms for inferring dynamical and microrheological quantities from individual random trajectories is discussed, with emphasis on a reliable interpretation of single-particle tracking experiments.