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...