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This paper puts forward a new generalized polynomial dimensionaldecomposition (PDD), referred to as GPDD, comprising hierarchically orderedmeasure-consistent multivariate orthogonal polynomials in dependent randomvariables. Unlike the existing PDD, which is valid strictly for independentrandom variables, no tensor-product structure is assumed or required. Importantmathematical properties of GPDD are studied by constructing dimension-wisedecomposition of polynomial spaces, deriving statistical properties of randomorthogonal polynomials, demonstrating completeness of orthogonal polynomialsfor prescribed assumptions, and proving mean-square convergence to the correctlimit, including when there are infinitely many random variables. The GPDDapproximation proposed should be effective in solving high-dimensionalstochastic problems subject to dependent variables.

Dimension-wise Multivariate Orthogonal Polynomials in General Probability Spaces