Uniform estimates for order statistics and Orlicz functions

Abstract

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ 1 , . . . , ξ n and a vector of scalarsterms of the values k and the Orlicz norm y x M of the vector y x = (1/x 1 , . . . , 1/x n ). Here M(t) is the appropriate Orlicz function associated 2 Y. Gordon et al. with the distribution function of the random variable |ξ 1 |, G(t) = P ({|ξ 1 | ≤ t}). For example, if ξ 1 is the standard N (0, 1) Gaussian random variable, then G(t) = 2 π t 0 e − s 2 2 ds and M(s) = 2 π s 0 e − 1 2t 2 dt.We would like to emphasize that our estimates do not depend on the length n of the sequence.

Probabilistic estimates for tensor products of random vectors

“…. , X n of a q-integrable random variable X are given (1 < q < p ≤ ∞), where |X| has a continuous density, then we obtain the following theorem in the flavor of the results in [9,1,17] (cf., Theorems 2.1, 2.5 below, and the discussion above). Theorem 1.2.…”

“…Building upon that, in the last decade these techniques initiated further research, were extended, and successfully used in several different areas of mathematics. Those include the local theory of Banach spaces, when studying symmetric subspaces of L 1 [20,22,23,18,15,16], probability theory, to obtain uniform estimates for order statistics [9] (see also [7,8,6,13]) as well as converse results on the distribution of random variables in connection with Musielak-Orlicz norms [1], or convex geometry, to obtain sharp bounds for several geometric functionals on random polytopes [2,4,3] such as the support function, the mean width and mean outer radii.…”

“…, X n be independent copies of an integrable random variable X. In [9,Lemma 5.2], it was proved that, if we define the Orlicz function M X by…”

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“…The following theorem was obtained in [10] and provides a formula for the Orlicz function M provided that we know the distribution of X:…”

On the Expectation of Operator Norms of Random Matrices

“…Note that in general the maximum of i.i.d. random variables weighted by coordinates of a vector a is equivalent to a certain Orlicz norm a M , where the function M depends only on the distribution of random variables (see [12,Corollary 2] and Lemma 5.2 in [11]).…”