Optimal and suboptimal quadratic forms for noncentered Gaussian processes

Denis S. Grebenkov
Phys. Rev. E 88, 032140 – Published 27 September 2013

Abstract

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 noncentered Gaussian processes, we construct the optimal quadratic form by using the spectral representation of 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.

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  • Received 30 June 2013

DOI:https://doi.org/10.1103/PhysRevE.88.032140

©2013 American Physical Society

Authors & Affiliations

Denis S. Grebenkov*

  • Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS–Ecole Polytechnique, 91128 Palaiseau, France

  • *denis.grebenkov@polytechnique.edu

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Vol. 88, Iss. 3 — September 2013

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