Orthogonal separation of independent Gaussian polynomial statistics
Let $n \ge 2$, let $X$ be a standard Gaussian vector in $\mathbb{R}^n$, and let $f, g \in \mathbb{R}[x_1, \dots, x_n]$ be nonconstant polynomials. Call $f$ and $g$ unlinked if there are an orthogonal matrix $O \in \mathbb{R}^{n \times n}$, an integer $1 \le k < n$ and polynomials $F \in \mathbb{R}[y_1, \dots, y_k]$ and $G \in \mathbb{R}[y_{k+1}, \dots, y_n]$ such that $$f(Oy) = F(y_1, \dots, y_k), \qquad g(Oy) = G(y_{k+1}, \dots, y_n) \qquad \text{for all } y \in \mathbb{R}^n.$$ If $f$ and $g$ are unlinked, then $f(X)$ and $g(X)$ are independent, since the coordinates of $O^{\mathsf T} X$ are independent. The converse holds for $n = 2$ (see Hong and Hu). Bhandari and Basu proved it for nonnegative convex polynomials with $f(0) = 0$; Hargé for convex functions $f, g$ square-integrable under the Gaussian measure, with $f$ real analytic and $f(x) \ge f(0)$ for all $x$; Hong and Hu for even quasi-convex polynomials; and Malicet, Nourdin, Peccati and Poly for polynomials of the form $\sum_{k=1}^{m} F_k^2$ with each $F_k$ in the $k$th Wiener chaos.
Problem. If $f(X)$ and $g(X)$ are independent, must $f$ and $g$ be unlinked?
Source: Kagan, Linnik and Rao (1973), the U-conjecture; see Hargé (2005) and Hong and Hu (2022).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- ST
- Keywords
- AIM Open Applied Problems, characterization problems, Gaussian vectors, independence
References
@book{kagan1973,
author = {Kagan, A. M. and Linnik, Yu. V. and Rao, C. Radhakrishna},
title = {Characterization Problems in Mathematical Statistics},
publisher = {Wiley},
address = {New York},
year = {1973},
}
@article{harge2005,
author = {Harg{\'e}, Gilles},
title = {Characterization of equality in the correlation inequality for convex functions, the {U}-conjecture},
journal = {Ann. Inst. H. Poincar{\'e} Probab. Statist.},
volume = {41},
number = {4},
year = {2005},
pages = {753--765},
doi = {10.1016/j.anihpb.2004.05.005},
}
@article{honghu2022,
author = {Hong, Hejing and Hu, Zechun},
title = {Unlinking theorem for symmetric quasi-convex polynomials},
journal = {Chinese J. Appl. Probab. Statist.},
volume = {38},
number = {1},
year = {2022},
pages = {151--158},
doi = {10.3969/j.issn.1001-4268.2022.01.011},
eprint = {2012.14568},
archivePrefix = {arXiv},
}
@article{bhandari2006,
author = {Bhandari, Subir Kumar and Basu, Ayanendranath},
title = {On the unlinking conjecture of independent polynomial functions},
journal = {J. Multivariate Anal.},
volume = {97},
number = {6},
year = {2006},
pages = {1355--1360},
doi = {10.1016/j.jmva.2005.06.007},
}
@article{malicet2016,
author = {Malicet, Dominique and Nourdin, Ivan and Peccati, Giovanni and Poly, Guillaume},
title = {Squared chaotic random variables: new moment inequalities with applications},
journal = {J. Funct. Anal.},
volume = {270},
number = {2},
year = {2016},
pages = {649--670},
doi = {10.1016/j.jfa.2015.10.013},
eprint = {1503.02154},
archivePrefix = {arXiv},
}
@misc{aim-gaussian-polynomial-unlinking,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Orthogonal separation of independent Gaussian polynomial statistics}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/317-gaussian-polynomial-unlinking.md},
note = {Entry 317 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00287, "Orthogonal separation of independent Gaussian polynomial statistics", clopen.solutions.