Evolutive instantaneous spectrum associated with partial autocorrelation function

Several approaches have been developed for the spectral analysis of nonstationary processes in the literature. Otherwise, it has been shown recently that, as in the stationary case, the partial autocorrelation function characterizes, like the autocovariance function, the second-order properties of the process. Our main result is the introduction of a new time-dependent power spectrum clearly related to this function. At each time, this spectrum describes a stationary situation in which the present is correlated with the past in the same way as our nonstationary process at this time. The properties of this spectrum are analysed. In particular, it is defined for all nonstationary processes and is in a one-to-one correspondence with the autocovariance function. Unfortunately, no spectral representation of the process is actually associated with it. This spectrum is also compared with two similar other spectra. Some examples of theoretical spectra and an estimated spectrum are considered for illustration.