Gap between orthogonal projectors—Application to stationary processes

In the statistics of processes, the usual convergence of projectors does not provide a good idea of the continuity of some phenomena. In order to address it, we introduce a measure of the gap between two projectors, that we link to another type of convergence of sequences of projectors. This allows to define the gap between two spectral measures and then to develop various properties associated with the latter. In particular, we establish the continuity of the convolution product. A study of the closeness between two continuous random functions and the exhibition of a common filter for them illustrate the possible applications in functional statistics.