Optimal and suboptimal quadratic forms for noncentered Gaussian processes.

Area of Science:

• Statistical physics
• Stochastic processes analysis

Background:

• Analyzing random trajectories of stochastic processes often involves quadratic forms like time-averaged mean square displacement (TA MSD) and velocity auto-correlation function (TA VACF).
• A narrow probability distribution of these quadratic forms is crucial for reducing statistical uncertainty in single measurements.

Purpose of the Study:

• To find an optimal quadratic form that minimizes a specific cumulant moment (e.g., variance) of the probability distribution.
• To establish a benchmark for the smallest achievable cumulant moment for discrete noncentered Gaussian processes.

Main Methods:

• Utilizing the spectral representation of cumulant moments for discrete noncentered Gaussian processes.
• Constructing the optimal quadratic form under the constraint of a fixed mean value.

Main Results:

• Derivation of a simple, explicit formula for the minimal achievable cumulant moment.
• Demonstration that this minimal moment can serve as a quality benchmark for other quadratic forms.
• Comparison of the optimal variance with variances of TA MSD and TA VACF for fractional Brownian motion with drift and noise.

Conclusions:

• The developed optimal quadratic form offers improved precision in analyzing stochastic process trajectories.
• The derived formula provides a valuable benchmark for assessing the efficiency of existing analytical methods.
• This work advances the understanding of statistical uncertainty reduction in single-trajectory analysis.