Finite product-law polynomial-chaos input data

Fix an integer $d\ge 1$, independent real random variables $Y_1,\dots ,Y_d$ with marginal Borel probability laws ${\rho }_1,\dots ,{\rho }_d$, and a degree cutoff vector $p=(p_1,\dots ,p_d)\in {\mathbb{N}}_0^d$. Their joint law is $\rho ={\rho }_1\otimes \cdots \otimes {\rho }_d$ on ${\mathbb{R}}^d$, as in Independent random elements have product joint law. For a Borel probability law ${\rho }_j$ on $\mathbb{R}$, define $\mathrm{supp}({\rho }_j):=\{x\in \mathbb{R}:{\rho }_j(U)>0\text{ for every open }U\ni x\}\mathrm{.}$ The complement is a union of zero-measure members of the countable rational interval basis, so countable additivity gives ${\rho }_j(\mathrm{supp}({\rho }_j))=1$. Set $\mathrm{\Gamma }={\prod }_{j=1}^d\mathrm{supp}({\rho }_j)$ and give it the trace Borel sigma-algebra from ${\mathbb{R}}^d$. The product rectangle formula gives $\rho (\mathrm{\Gamma })=1$. Whenever the measure space is written as $(\mathrm{\Gamma },\mathcal{B}(\mathrm{\Gamma }),\rho )$, $\rho$ denotes the restriction of the product law on ${\mathbb{R}}^d$ to $\mathcal{B}(\mathrm{\Gamma })$; it is therefore a probability measure. The product rectangle formula also identifies this restriction with the finite product of the marginal laws restricted to their supports. The cutoff vector $p$ is admissible when, for every $j=1,\dots ,d$, $\int \mathrm{\mid }y{\mathrm{\mid }}^{2p_j}d{\rho }_j(y)<\mathrm{\infty }$ and the monomial Gram matrix $G_j=(\int y^{r+s}d{\rho }_j(y){)}_{0\le r,s\le p_j}$ is positive definite, meaning $c^T{G}_{j}c>0$ for each nonzero $c\in {\mathbb{R}}^{p_j+1}$. Throughout the pair, assume the selected cutoff vector is admissible. The admissible degree box is $\{\alpha \in {\mathbb{N}}_0^d:0\le {\alpha }_j\le p_j\text{ for every }j\}$. Admissibility gives a finite inner product on each stated polynomial space. It is only a finite-degree condition; it does not assert that polynomials are dense in any $L^2({\rho }_j)$.