Abstract
In several different aspects of algebra and topology the following problem is of interest: find the maximal number of unitary antisymmetric operators Ui
in with the property UiUj
= –UjUi
(i ≠ j). The solution of this problem is given by the Hurwitz-Radon-Eckmann formula. We generalize this formula in two directions: all the operators Ui
must commute with a given arbitrary self-adjoint operator and H can be infinite-dimensional. Our second main result deals with the conditions for almost sure orthogonality of two random vectors taking values in a finite or infinite-dimensional Hilbert space H. Finally, both results are used to get the formula for the maximal number of pairwise almost surely orthogonal random vectors in H with the same covariance operator and each pair having a linear support in H ⊕ H.
The paper is based on the results obtained jointly with N.P.Kandelaki (see [Vakhania and Kandelaki, Dokl. Akad. Nauk SSSR 294: 528-531, 1987, Dokl. Akad. Nauk SSSR 296: 265-266, 1988, Trudy Inst. Vychisl. Mat. Akad. Nauk Gruz. SSR 28: 11-37, 1988]).
Limiting Behavior of the Partial Sum for Negatively Superadditive Dependent Random Vectors in Hilbert Space
