Time-averaged quadratic functionals of a Gaussian process

The characterization of a stochastic process from its single random realization is a challenging problem for most single-particle tracking techniques which survey an individual trajectory of a tracer in a complex or viscoelastic medium. We consider two quadratic functionals of the trajectory: the time-averaged mean-square displacement (MSD) and the time-averaged squared root mean-square displacement (SRMS). For a large class of stochastic processes governed by the generalized Langevin equation with arbitrary frictional memory kernel and harmonic potential, the exact formulas for the mean and covariance of these functionals are derived. The formula for the mean value can be directly used for fitting experimental data, e.g., in optical tweezers microrheology. The formula for the variance (and covariance) allows one to estimate the intrinsic fluctuations of measured (or simulated) time-averaged MSD or SRMS for choosing the experimental setup appropriately. We show that the time-averaged SRMS has smaller fluctuations than the time-averaged MSD, in spite of much broader applications of the latter one. The theoretical results are successfully confirmed by Monte Carlo simulations of the Langevin dynamics. We conclude that the use of the time-averaged SRMS would result in a more accurate statistical analysis of individual trajectories and more reliable interpretation of experimental data.