Local stationarity of L2(ℝ) processes

This paper shows how the sampling theorem relates with the variations along time of the second order statistics of L²(ℝ) nonstationary processes. As a consequence, and mainly due to the positive semidefiniteness of autocorrelation functions, it is possible to conclude if a nonstationary process is locally stationary (i.e., if its second order statistics vary slowly along time) by the direct observation of its 2-dimension power spectrum or its Wigner distribution. A simple example illustrates how two different strategies for the estimation of autocorrelation functions from a small number of data can lead to opposite results in terms of local stationarity.