PreprintCC BY 4.0arXiv
Two short proofs of incompatibility for correlation matrices
Abstract
Let Pd be the elliptope of d × d correlation matrices, and, for a standardised law F, let SdF be the set of correlation matrices of random vectors with all margins F. We give two short proofs. First, an eleven-term representation of |x|4 as a sum of fourth powers of linear forms on ℝ4 yields S11 ≠ P11 for uniform margins. Second, the six diagonals of the regular icosahedron yield S6F ≠ P6 for arcsine margins. Together with a result of Devroye and Letac, this gives SdF = Pd if and only if d ≤ 5, settling a conjecture of theirs.
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BibTeX
@misc{phillips2026short,
title = {{Two short proofs of incompatibility for correlation matrices}},
author = {Cedric Phillips},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14610},
eprint = {2609.14610},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14610},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two short proofs of incompatibility for correlation matrices
AU - Cedric Phillips
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14610
UR - https://arxiv.org/abs/2609.14610
AB - Let P_d be the elliptope of d × d correlation matrices, and, for a standardised law F, let S_d^F be the set of correlation matrices of random vectors with all margins F. We give two short proofs. First, an eleven-term representation of |x|^4 as a sum of fourth powers of linear forms on ℝ^4 yields S_11 ≠ P_11 for uniform margins. Second, the six diagonals of the regular icosahedron yield S_6^F ≠ P_6 for arcsine margins. Together with a result of Devroye and Letac, this gives S_d^F = P_d if and only if d ≤ 5, settling a conjecture of theirs.
ER -