  This paper introduces a new generalized polynomial chaos expansion (PCE)
comprising measure-consistent multivariate orthonormal polynomials in dependent
random variables. Unlike existing PCEs, whether classical or generalized, no
tensor-product structure is assumed or required. Important mathematical
properties of the generalized PCE are studied by constructing orthogonal
decomposition of polynomial spaces, explaining completeness of orthogonal
polynomials for prescribed assumptions, exploiting whitening transformation for
generating orthonormal polynomial bases, and demonstrating mean-square
convergence to the correct limit. Analytical formulae are proposed to calculate
the mean and variance of a truncated generalized PCE for a general output
variable in terms of the expansion coefficients. An example derived from a
stochastic boundary-value problem illustrates the generalized PCE approximation
in estimating the statistical properties of an output variable for 12 distinct
non-product-type probability measures of input variables.
