A CLT in Stein's distance for generalized Wishart matrices and higher order tensors
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We study the convergence along the central limit theorem for sums of independent tensor powers, 1/√(d)∑_i=1^d X_i^⊗ p. We focus on the high-dimensional regime where X_i ∈R^n and n may scale with d. Our main result is a proposed threshold for convergence. Specifically, we show that, under some regularity assumption, if n^2p-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 X_i^⊗ p.
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