# Order statistics of vectors with dependent coordinates, and the Karhunen–Loève basis

@article{Litvak2018OrderSO,
title={Order statistics of vectors with dependent coordinates, and the Karhunen–Lo{\e}ve basis},
author={Alexander E. Litvak and Konstantin E. Tikhomirov},
journal={The Annals of Applied Probability},
year={2018}
}`
• Published 7 September 2016
• Mathematics
• The Annals of Applied Probability
Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not identically distributed coordinates and let $T$ be an orthogonal trasformation of $\mathbb R^n$. We show that the random vector $Y=T(X)$ satisfies $$\mathbb E\sum\limits_{j=1}^k j\mbox{-}\min_{i\leq n}{X_{i}}^2 \leq C\mathbb E\sum\limits_{j=1}^k j\mbox{-}\min_{i\leq n}{Y_{i}}^2$$ for all $k 0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni… Expand
3 Citations
Estimates for Order Statistics in Terms of Quantiles
• Mathematics
• Journal of Mathematical Sciences
• 2019
Let X1, . . .,Xn be independent nonnegative random variables with cumulative distribution functions F1, F2, . . . , Fn satisfying certain (rather mild) conditions. We show that the median of kthExpand
Around the Simplex Mean Width Conjecture
In this note we discuss an old conjecture in Convex Geometry asserting that the regular simplex has the largest mean width among all simplices inscribed into the Euclidean ball and its relation toExpand
Compressibility of Network Opinion and Spread States in the Laplacian-Eigenvector Basis
• Computer Science, Engineering
• ArXiv
• 2021
The three case studies indicate that state snapshots from opinion-evolution and spread processes allow terse representations, which nevertheless capture their rich propagative dynamics. Expand

#### References

SHOWING 1-10 OF 25 REFERENCES
Uniform estimates for order statistics and Orlicz functions
• Mathematics
• 2008
We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ1, … , ξn and a vector of scalars x = (x1, … , xn), and 1 ≤ k ≤ n, weExpand
Tail estimates for norms of sums of log‐concave random vectors
• Mathematics
• 2014
We establish new tail estimates for order statistics and for the Euclidean norms of projections of an isotropic log-concave random vector. More generally, we prove tail estimates for the norms ofExpand
Minima of sequences of Gaussian random variables
• Mathematics
• 2005
Abstract For a given sequence of real numbers a 1 , … , a n we denote the k-th smallest one by k - min 1 ⩽ i ⩽ n a i . We show that there exist two absolute positive constants c and C such that forExpand
On the minimum of several random variables
• Mathematics
• 2006
For a given sequence of real numbers a 1 ,..,a n , we denote the kth smallest one by k-min 1<i<nai . Let A be a class of random variables satisfying certain distribution conditions (the classExpand
Orlicz norms of sequences of random variables
• Mathematics
• 2002
Let f i , i = 1,...,n, be copies of a random variable f and let N be an Orlicz function. We show that for every x ∈ R n the expectation E||(x i f i ) n i=1 || N is maximal (up to an absoluteExpand
A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition
The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications. Expand
Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables
LetX1,X2, …,Xn be a sequence of independent random variables, letM be a rearrangement invariant space on the underlying probability space, and letN be a symmetric sequence space. This paper gives anExpand
Random aspects of high-dimensional convex bodies
• Mathematics
• 2000
In this paper we study geometry of compact, not necessarily centrally symmetric, convex bodies in R. Over the years, local theory of Banach spaces developed many sophisticated methods to studyExpand
Diameters of sections and coverings of convex bodies
• Mathematics
• 2006
Abstract We study the diameters of sections of convex bodies in R N determined by a random N × n matrix Γ , either as kernels of Γ * or as images of Γ . Entries of Γ are independent random variablesExpand
EXTREMAL PROPERTIES OF ORTHOGONAL PARALLELEPIPEDS AND THEIR APPLICATIONS TO THE GEOMETRY OF BANACH SPACES
It is proved that the distribution function for the maximum of the modulus of a set of jointly Gaussian random variables with given variance and zero mean is minimal if these variables areExpand