Should we always trust in the autocorrelation function?

The autocorrelation function and, in general, the cross-correlation function is often misused. Auto-covariance and the related spectrum methods all assume at least a wide-sense stationarity. In practice, many signals (especially biomedical data) show significant nonlinear and non-gaussian characteristics, such as the presence of nonlinear effects of phase coupling among the signal frequency components. Instead, the methods based on spectral analysis fail to properly deal with the nonlinearity and non-gaussianity of the processes. 

Thus, stationarity, linearity and gaussianity should be previously assessed (e.g., by unit-root test and Hinich test for linearity) . It is known that shocks to a unit root process have permanent effects which do not decay as they would if the process were stationary, that is, a unit root process has a variance that depends on t, and diverges to infinity. In this occurrence, the series can be differenced to render it stationary. 

Furthermore, for gaussian distribution, stationarity tells us that different frequency components are statistically independent. It implies that if the process is gaussian then the representation in principal components of the original data set is stochastically independent. In fact, through the Wiener-Khintchine theorem, the auto-correlation function is the Fourier transform of the spectrum. 

Although the values of a gaussian process at different times are not independent, the different frequencies components are independent. Hence, the frequency components provide the coordinate system in which the covariance matrix is diagonal (principal components) and the power spectrum measures the variance of these independent variables. 
Thus, the power spectrum, which is a function of frequency, generalizes the concept of variance to the case of time dependent signals. 

Amazingly, different mechanisms may produce the same power spectrum from qualitatively different data. Then, it would be relevant to study the nonlinearity of the variability. Linear variability occurs if the phases at the different frequencies in the Fourier spectrum of a time series are uncorrelated with one another, while time series with non-linear variability show Fourier spectra with correlations between the phases at different frequencies. Some particularly useful tools for studying non-linearity are the bispectrum and the closely related bicoherence. 

In the case of a linear random process, the bicoherence is a mere constant independent of the frequency. This constant is zero if the random process is a gaussian random process. In this case, both the bispectrum and bicoherence are zero. 
If the signal is a linear random process, then the phases are also randomly distributed. Consequently, the power spectrum and so the auto-correlation function, contains all the information (sufficience) of the original signal.