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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2604.10323 |
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| _version_ | 1866918440833384448 |
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| author | Castillo, Isaac Pérez Leyvraz, François Quintanar, Miguel Eduardo Gómez Ballesteros, Andrés Álvarez |
| author_facet | Castillo, Isaac Pérez Leyvraz, François Quintanar, Miguel Eduardo Gómez Ballesteros, Andrés Álvarez |
| contents | 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 $\{ω_i\}$, we obtain an exact characterization of the joint law of the finite-time estimators $\{S(ω_i,T)\}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10323 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Beyond Whittle: exact finite-time multispectral statistics from a single Brownian trajectory in a harmonic trap Castillo, Isaac Pérez Leyvraz, François Quintanar, Miguel Eduardo Gómez Ballesteros, Andrés Álvarez Statistical Mechanics 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 $\{ω_i\}$, we obtain an exact characterization of the joint law of the finite-time estimators $\{S(ω_i,T)\}$, 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. |
| title | Beyond Whittle: exact finite-time multispectral statistics from a single Brownian trajectory in a harmonic trap |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2604.10323 |