A sampling theorem for multivariate stationary processes

The notion of sampling for second-order q -variate processes is defined. It is shown that if the components of a q -variate process (not necessarily stationary) admits a sampling theorem with some sample spacing, then the process itself admits a sampling theorem with the same sample spacing. A sampling theorem for q -variate stationary processes, under a periodicity condition on the range of the spectral measure of the process, is proved in the spirit of Lloy's work. This sampling theorem is used to show that if a q -variate stationary process admits a sampling theorem, then each of its components will admit a sampling theorem too.