  We study the statistical properties of the time-averaged mean-square
displacements (TAMSD). This is a standard non-local quadratic functional for
inferring the diffusion coefficient from an individual random trajectory of a
diffusing tracer in single-particle tracking experiments. For Brownian motion,
we derive an exact formula for the Laplace transform of the probability density
of the TAMSD by mapping the original problem onto chains of coupled harmonic
oscillators. From this formula, we deduce the first four cumulant moments of
the TAMSD, the asymptotic behavior of the probability density and its accurate
approximation by a generalized Gamma distribution.
