One of the popular methods for numerical solution of partial differential equations with data dependent on random parameters is the stochastic Galerkin method. Approximation spaces of this method are usually tensor products of finite elements of spatial variables and of sets of orthogonal polynomials of random variables. After discretization, obtained sets of linear equations are of a huge dimensionality and any kind of reduction or preconditioning is desired.
We study a natural hierarchical structure of orthogonal polynomials of random variables, both for complete polynomials and for tensor products of polynomials of individual random variables. For some splittings of these spaces upper bounds to the strengthened Cauchy-Bunyakowski-Schwarz (CBS) constants with respect to energy scalar product can be derived. Small CBS constants lead to efficient two-by-two block preconditioning and to reliable a posteriori error estimates as well. We introduce sharp upper bounds to the CBS constants considering uniform or normal distributions of random variables. Introduced numerical examples confirm these results.