Random vectors with values in quaternion Hilbert spaces

The paper is devoted to a systematic study of basic primary concepts and facts that could be regarded as a part of the apparatus of a future theory of quaternion random variables and vectors. Mainly we deal with the infinite-dimensional case. Defined and analyzed are basic concepts of the theory such as mathematical expectation, covariance and cross-covariance operators, characteristic functional, Gaussian measures–for random vectors with values in separable Hilbert space over the field of quaternions.
The paper is self-contained. However, conceptually it can be regarded as a natural continuation of the work of N. N. Vakhania and N. P. Kandelaki [Theory Probab. Appl., 41 (1996), pp. 116–131], in which random vectors with values in complex Hilbert spaces are considered; the organization of this paper is similar to that earlier work.
Despite an apparent similarity in the formulations, noncommutativity in the quaternion case brings in a specific peculiarity, often hidden and unexpected. Indeed, to overcome these difficulties caused by noncommutativity one requires not so much ingenuity as thoroughness and accuracy in giving definitions and formulating and proving results.