Groupe d'opérateurs unitaires déduit d'une mesure spectrale – une application

Abstract

In this Note, given the stationary continuous random function ( X t ) t ∈ R and a pair ( a , b ) of reals we will define all the stationary processes ( X ( t 1 , t 2 ) ′ ) ( t 1 , t 2 ) ∈ R × R such that ( X ( b t , a t ) ′ ) t ∈ R = ( X t ) t ∈ R . For this purpose, we use the product of spectral measure convolution, as defined in [A. Boudou, Y. Romain, On spectral and random measures associated to discrete and continuous-time processes, Statist. Probab. Lett. 59 (2002) 145–157], and we develop other mathematical tools which offer a wider interest than our initial preoccupation. So, we use the concept of transposed homomorphism, which allows us to evenly define the concept of a group of unitary operators deduced from a spectral measure. To cite this article: A. Boudou, C. R. Acad. Sci. Paris, Ser. I 344 (2007).