6533b863fe1ef96bd12c79d5

RESEARCH PRODUCT

The variance of the ℓnp-norm of the Gaussian vector, and Dvoretzky's theorem

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Let n be a large integer, and let G be the standard Gaussian vector in Rn. Paouris, Valettas and Zinn (2015) showed that for all p∈[1,clogn], the variance of the ℓnp-norm of G is equivalent, up to a constant multiple, to 2ppn2/p−1, and for p∈[Clogn,∞], to (logn)−1. Here, C,c>0 are universal constants. That result left open the question of estimating the variance for p logarithmic in n. In this paper, the question is resolved by providing a complete characterization of Var∥G∥p for all p. It is shown that there exist two transition points (windows) in which the behavior of Var∥G∥p changes significantly. Some implications of the results are discussed in the context of random Dvoretzky's theorem for ℓnp.