Quaternion Gaussian Random Variables

The main result of this paper is the formulation and proof of Polya's characterization theorem of Gaussian random variables with values in quaternion algebras. We begin with some preliminary information about quaternions and quaternion random variables. More complete information about quaternions (except the items connected with the jointly quaternion systems) can be found in [N. N. Vakhania, Theory Probab. Appl., 43 (1999), pp. 99–115]. Closely related to the present paper is also the paper [N. N. Vakhania, J. Complexity, 13 (1997), pp. 480–488] in which Polya's characterization theorem for complex random variables is proved.