Dimensionwise multivariate orthogonal polynomials in general probability spaces

Abstract

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, encompassing hierarchically organized, measure-consistent multivariate orthogonal polynomials in dependent random variables. In contrast to the existing PDD, which is valid strictly for independent random variables, no tensor-product structure or product-type probability measure is imposed or necessary. Fundamental mathematical properties of GPDD are examined by creating dimensionwise decomposition of polynomial spaces, deriving statistical properties of random orthogonal polynomials, demonstrating completeness of orthogonal polynomials for requisite assumptions, and upholding mean-square convergence to the correct limit. The GPDD approximation proposed should be effective in solving high-dimensional stochastic problems subject to dependent variables with a large class of non-product-type probability measures.