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 $\left\{{\omega }_i\right\}$, we obtain an exact characterization of the joint law of the finite-time estimators $\left\{S({\omega }_i,T)\right\}$, together with a covariance-explicit Gaussian representation for the associated Fourier projections. This representation makes the observation-window-induced interfrequency 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.

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• Brownian motion
• Stochastic inference
• Langevin equation
• Time series analysis