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Recent advances in light microscopy have spawned new research frontiers inmicrobiology by working around the diffraction barrier and allowing for theobservation of nanometric biological structures. Microrheology is the study ofthe properties of complex fluids, such as those found in biology, through thedynamics of small embedded particles, typically latex beads. Statistics basedon the recorded sample paths are then used by biophysicists to inferrheological properties of the fluid. In the biophysical literature, the mainstatistic for characterizing diffusivity is the so-named mean squareddisplacement (MSD) of the tracer particles. Notwithstanding the central roleplayed by the MSD, its asymptotic distribution in different cases has not yetbeen established. In this paper, we tackle this problem. We take a pathwiseapproach and assume that the particle movement undergoes a Gaussian,stationary-increment stochastic process. We show that as the sample and theincrement lag sizes go to infinity, the MSD displays Gaussian or non-Gaussianlimiting distributions, as well as distinct convergence rates, depending on thediffusion exponent parameter. We illustrate our results analytically andcomputationally based on fractional Brownian motion and the (integrated)fractional Ornstein-Uhlenbeck process.

The asymptotic distribution of the pathwise mean squared displacement in single particle tracking experiments