Статья «THE VARIANCE OF THE fJI-NORM OF THE GAUSSIAN VECTOR, AND DVORETZKY'S THEOREM, "Алгебра и анализ"»

Let n be a large integer, and let G be the standard Gaussian vector in R. Paouris, Valettas and Zinn (2015) showed that for all p € [1, clogn], the variance of the ^-norm of G is equivalent, up to a constant multiple, to ^·п', and for p € [Clogn, то], to (log n). 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|| for all p. It is shown that there exist two transition points (windows) in which the behavior of Var||Cr||p changes significantly. Some implications of the results are discussed in the context of random Dvoretzky's theorem for ?'p.