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Random Matrix Theory / ProbabilityResearchAI-Generated

Does the Marchenko-Pastur law hold for tensor powers of exchangeable vectors that are not unconditional, and if so under what moment or dependence conditions?

Related: Marchenko-Pastur theorem for independent entries, Wigner semicircle law universality, de Finetti theorem for exchangeable sequences

The Marchenko-Pastur law describes the limiting spectral distribution of sample covariance matrices formed from high-dimensional data. The paper under consideration establishes this law for tensor powers of exchangeable unconditional vectors, where unconditionality means the distribution is invariant under sign flips of coordinates. The open problem is whether this universality result extends to exchangeable vectors that lack the unconditionality assumption, and precisely which structural conditions on the dependence between coordinates are necessary and sufficient for the Marchenko-Pastur law to persist in the tensor power regime.

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