  Recent advances in light microscopy have spawned new research frontiers in
microbiology by working around the diffraction barrier and allowing for the
observation of nanometric biological structures. Microrheology is the study of
the properties of complex fluids, such as those found in biology, through the
dynamics of small embedded particles, typically latex beads. Statistics based
on the recorded sample paths are then used by biophysicists to infer
rheological properties of the fluid. In the biophysical literature, the main
statistic for characterizing diffusivity is the so-named mean squared
displacement (MSD) of the tracer particles. Notwithstanding the central role
played by the MSD, its asymptotic distribution in different cases has not yet
been established. In this paper, we tackle this problem. We take a pathwise
approach and assume that the particle movement undergoes a Gaussian,
stationary-increment stochastic process. We show that as the sample and the
increment lag sizes go to infinity, the MSD displays Gaussian or non-Gaussian
limiting distributions, as well as distinct convergence rates, depending on the
diffusion exponent parameter. We illustrate our results analytically and
computationally based on fractional Brownian motion and the (integrated)
fractional Ornstein-Uhlenbeck process.
