Polynomial-Gaussian vectors and polynomial-Gaussian processes (Q2732355)

Let \({\mathbf X}_d=(X_1,\dots, X_d)\) denote a \(d\)-dimensional r.v., the density of which is a product of a nonnegative polynomial in \(x_1\) and a \(d\)-dimensional Gaussian density. \({\mathbf X}_d\) has a \(d\)-dimensional polynomial-Gaussian distribution \((\text{PGD}_d)\). The characteristic function and the first and the second moments of \((\text{PGD}_d)\) r.v.s are found. Every \(\delta\)-dimensional \((\delta<d)\) marginal distribution of \((\text{PGD}_d)\) is \((\text{PGD}_\delta)\). Properties of sums of \((\text{PGD}_d)\) r.v.s are studied. A characterization of \((\text{PGD}_d)\) is given. A stochastic process is constructed such that its one-dimensional distributions are \((\text{PGD}_1)\). NEWLINENEWLINENEWLINESee also \textit{M. Evans} and \textit{T. Swartz} [Commun. Stat., Theory Methods 23, No. 4, 1123-1248 (1994)] and \textit{A. Plucińska} [Demonstr. Math. 32, No. 1, 195-206 (1999; Zbl 0932.60014)].