| Summary: | We study the a central limit theorem for sums of independent tensor powers, 1/√d Σi=1d Xi⊗p. We focus on the high-dimensional regime where Xi ∈ ℝn and n may scale with d. Our main result is a proposed threshold for convergence. Specifically, we show that, under some regularity assumption, if n2p-1 ≪ d, then the normalized sum converges to a Gaussian. The results apply, among others, to symmetric uniform log-concave measures and to product measures. This generalizes several results found in the literature. Our main technique is a novel application of optimal transport to Stein's method, which accounts for the low-dimensional structure, which is inherent in Xi⊗p. © 2021 The Author(s).
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