On proximity between PCA in the frequency domain and usual PCA

The principal components analysis (PCA) in the frequency domain of a stationary p-dimensional time series (X _n ) _n∈? leads to a summarizing time series written as a linear combination series X′ _n =∑ _m C _m ° X _n?m . Therefore, we observe that, when the coefficients C _m , m≠0, are close to 0, this PCA is close to the usual PCA, that is the PCA in the temporal domain. When the coefficients tend to 0, the corresponding limit is said to satisfy a property noted 𝒫, of which we will study the consequences. Finally, we will examine, for any series, the proximity between the two PCAs.