Exact finite-time multispectral statistics from a single Brownian trajectory in a harmonic trap

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 interfrequency correlations explicit and shows how they vanish as T→∞, 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.