Absolute continuity of two-dimensional polynomial random vectors

Abstract

Let $X=\{X_j{\}}_{j=1}^{\infty }$ be a sequence of independent random variables whose densities and moments of order $2d$ are uniformly bounded. For a random vector $f(X)=(f_1(X),f_2(X))$ whose components are polynomial functionals of degree at most $d$, we prove that \[ [[f]]_{μ,\infty}^{\frac1{2d-1}}μ(f\in A) \le C\bigl(λ_2(A)\bigr)^{\frac1{2d-1}} \] for every Borel set $A\subset {\mathbb{R}}^2$, where $C$ depends only on $d$ and the uniform density and moment bounds, and ${\lambda }_2$ denotes the Lebesgue measure on ${\mathbb{R}}^2$. Here $[[f]{]}_{\mu ,\infty }$ measures the failure of proportionality of the highest-order orthogonal-chaos components of $f_1$ and $f_2$ with respect to the law $\mu$ of $X$. Consequently, whenever these components are not proportional, the law of $f$ admits a density in the weak Lorentz space $L^{\frac{2d-1}{2d-2},\infty }({\mathbb{R}}^2)$. This recovers the dichotomy established by Nualart and Tudor for two-dimensional Wiener chaos vectors and extends it beyond the Gaussian setting. We also obtain the lower bound \[ \int_{\mathbb R^\infty}Δ_f\,dμ\ge C[[f]]_{μ,\infty}^2, \] where ${\Delta }_f$ is the determinant of the Gram matrix of $\nabla f_1$ and $\nabla f_2$. In the special case of Gaussian measures, this gives a relaxed version of the estimate conjectured by Nourdin, Nualart, and Poly.