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

TL;DR

This paper establishes the asymptotic distribution of the mean squared displacement in single particle tracking, revealing Gaussian or non-Gaussian limits depending on the diffusion exponent, with implications for biophysical analysis.

Contribution

It provides the first theoretical characterization of the asymptotic distribution of the MSD in various diffusion regimes for Gaussian stationary-increment processes.

Findings

MSD converges to Gaussian or non-Gaussian limits depending on the diffusion exponent.

Different convergence rates are observed based on the diffusion regime.

Results are demonstrated analytically and through simulations with fractional Brownian motion and fractional Ornstein-Uhlenbeck processes.

Abstract

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.