Article

Bounds on the concentration function in terms of Diophantine approximation

School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Comptes Rendus Mathematique (Impact Factor: 0.43). 07/2007; DOI: 10.1016/j.crma.2007.10.006
Source: arXiv

ABSTRACT We demonstrate a simple analytic argument that may be used to bound the Levy concentration function of a sum of independent random variables. The main application is a version of a recent inequality due to Rudelson and Vershynin, and its multidimensional generalisation.

0 Bookmarks
 · 
70 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The paper deals with studying a connection of the Littlewood-Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of results of Arak (1980) are given. They show a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random vectors for arbitrary distributions of summands.
    11/2014;
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator ${S : \ell_p^n \to \mathbb{R}^N}$ such that for every 0 < r < p < 2 with r ≤ 1, the operator ${S_r = S : \ell_p^n \to \ell_r^N}$ satisfies with overwhelming probability that ${\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}$ , where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.
    Mathematische Annalen 01/2011; 350(4). · 1.20 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: The paper is devoted to “proportional” embeddings of ℓ p n into ℓ r N . A typical result in this direction is the B. S. Kashin [Math. USSR, Izv. 11, 317–333 (1977; Zbl 0378.46027)] theorem: For any η>0, for any n, ℓ 2 n ↪ cℓ 1 N , where N=(1+η)n and the constant of isomorphism c depends only on η. All results of this sort are random: one constructs a random operator from ℓ p n into ℓ r N and proves that with a positive probability this operator has “nice” constant of isomorphism. The authors give, for any 0<p<2 and any natural numbers n<N, an explicit definition of a random operator S:ℓ p n →ℝ N with the following property. For every 0<r<p, r≤1, the operator S r =S:ℓ p n →ℓ r N satisfies with overwhelming probability that ∥S r ∥∥S r -1 ∥≤c n/(N-n) , where c>0 depends only on p and r. The authors note that these operators S r have already been defined in [G. Pisier, Trans. Am. Math. Soc., 276, 201–211 (1983; Zbl 0509.46016)] for the almost isometric result.
    Mathematische Annalen 01/2011; 350(4). · 1.20 Impact Factor

Full-text

Download
0 Downloads
Available from