Multiple stochastic signals possess inherent statistical correlations, yet conventional sampling methods that process each channel independently result in data redundancy. To leverage this correlation for efficient sampling, we model correlated channels as a linear combination of a smaller set of uncorrelated, wide-sense stationary latent sources. We establish a theoretical lower bound on the total sampling density for zero mean-square error reconstruction, proving it equals the ratio of the joint spectral bandwidth of latent sources to the number of correlated signal channels. We then develop a constructive multi-band sampling scheme that attains this bound. The proposed method operates via spectral partitioning of the latent sources, followed by spatio-temporal sampling and interpolation. Experiments on synthetic and real datasets confirm that our scheme achieves near-lossless reconstruction precisely at the theoretical sampling density, validating its efficiency.