Beyond Whittle: exact finite-time multispectral statistics from a single Brownian trajectory in a harmonic trap

TL;DR

This paper develops an exact finite-time multispectral theory for a Brownian particle in a harmonic trap, enabling precise spectral inference from single trajectories and quantifying finite-time effects.

Contribution

It introduces a finite-time joint law of spectral estimators, explicit inter-frequency correlations, and a hierarchy of spectral likelihoods for improved inference from single trajectories.

Findings

Monte Carlo simulations validate the finite-time theory.

Finite-time correlations affect single-trajectory spectral estimates.

The theory provides a benchmark beyond asymptotic approximations.

Abstract

Power spectral densities are often interpreted through ensemble averages and long-time asymptotics. In many experiments, however, only a single finite record is available, so spectral estimators remain broadly distributed and the usual independence assumptions across frequencies need not hold. Here we develop an exact finite-$T$ multispectral theory for an overdamped Brownian particle in a harmonic trap. For a collection of frequencies $\{\omega_i\}$, we obtain an exact characterization of the joint law of the finite-time estimators $\{S(\omega_i,T)\}$, together with a covariance-explicit Gaussian representation for the associated Fourier projections. This representation makes the observation-window-induced inter-frequency correlations explicit and shows how they vanish as $T\to\infty$, thereby recovering the asymptotic Whittle picture. We then use this structure to formulate a hierarchy of spectral likelihoods for inference from a single trajectory, ranging from the factorized Whittle approximation to blockwise covariance-aware approximations in frequency space. Monte Carlo simulations validate the finite-time theory and quantify the effect of neglected cross-frequency correlations on single-trajectory estimates of the trap parameters. Our results provide a controlled finite-time benchmark for spectral inference beyond the asymptotic regime.