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This paper introduces a new generalized polynomial chaos expansion (PCE)comprising multivariate Hermite orthogonal polynomials in dependent Gaussianrandom variables. The second-moment properties of Hermite polynomials reveal aweakly orthogonal system when obtained for a general Gaussian probabilitymeasure. Still, the exponential integrability of norm allows the Hermitepolynomials to constitute a complete set and hence a basis in a Hilbert space.The completeness is vitally important for the convergence of the generalizedPCE to the correct limit. The optimality of the generalized PCE and theapproximation quality due to truncation are discussed. New analytical formulaeare proposed to calculate the mean and variance of a generalized PCEapproximation of a general output variable in terms of the expansioncoefficients and statistical properties of Hermite polynomials. However, unlikein the classical PCE, calculating the coefficients of the generalized PCErequires solving a coupled system of linear equations. Besides, the varianceformula of the generalized PCE contains additional terms due to statisticaldependence among Gaussian variables. The additional terms vanish when theGaussian variables are statistically independent, reverting the generalized PCEto the classical PCE. Numerical examples illustrate the generalized PCEapproximation in estimating the statistical properties of various outputvariables.

Wiener-Hermite Polynomial Expansion for Multivariate Gaussian Probability Measures