Random vectors with values in complex Hilbert spaces

An attempt is made to give a more or less systematic approach to the study of some basic probability concepts related to random vectors with values in a complex Hilbert space. In a natural way, one can associate with such a random vector a pair of random vectors with values in a relevant real Hilbert space, and this yields two possible approaches to describe complex random vectors. These two approaches are actually the same with respect to the measurability problem and the concept of mathematical expectation. The situation is different, however, with regard to problems connected with a covariance operator. The main concept of the paper is a proper random vector the one for which these two approaches give the same results for problems related to covariance operator as well. We study the circle of problems connected with the concept of proper random vectors.