  Individual random trajectories of stochastic processes are often analyzed by
using quadratic forms such as time averaged (TA) mean square displacement (MSD)
or velocity auto-correlation function (VACF). The appropriate quadratic form is
expected to have a narrow probability distribution in order to reduce
statistical uncertainty of a single measurement. We consider the problem of
finding the optimal quadratic form that minimizes a chosen cumulant moment
(e.g., the variance) of the probability distribution, under the constraint of
fixed mean value. For discrete non-centered Gaussian processes, we construct
the optimal quadratic form by using the spectral representation of the cumulant
moments. Moreover, we obtain a simple explicit formula for the smallest
achievable cumulant moment that may serve as a quality benchmark for other
quadratic forms. We illustrate the optimality issues by comparing the optimal
variance with the variances of the TA MSD and TA VACF of fractional Brownian
motion superimposed with a constant drift and independent Gaussian noise.
