The discrete Fourier transform (DFT) and the discrete cosine transform (DCT) both decompose a signal into its frequency-domain spectrum. One property that I have seen praised across various domains such as image processing, audio/speech processing and more, is that it tends to produce decorrelated coefficients. For example, subsequent coefficients of a signal might tend to be statistically correlated, but its Fourier- or cosine-transformed counterpart does not exhibit such behaviour. As a result, the statistical properties of the latter (e.g. covariance matrix) can be modeled much more efficiently using diagonal matrices instead of dense matrices, among other benefits.
However, I do not immediately see why this decorrelation should hold true in general. What are the reasons behind this fact? Where is the connection between the DFT or DCT's frequency-decomposing property and the described decorrelating property?