• Corpus ID: 231925280

Anti-concentration of random variables from zero-free regions

  title={Anti-concentration of random variables from zero-free regions},
  author={Marcus Michelen and Julian Sahasrabudhe},
: This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let X be a random variable taking values in { 0 ,..., n } with P ( X = 0 ) P ( X = n ) > 0 and with probability generating function f X . We show that if all of the zeros ζ of f X satisfy | arg ( ζ ) | (cid:62) δ and R − 1 (cid:54) | ζ | (cid:54) R then Var ( X ) (cid:62) cR − 2 π / δ n , where c > 0 is a absolute constant. We show that this… 



Inverse Littlewood-Offord theorems and the condition number of random discrete matrices

Consider a random sum r)\V\ + • • • + r]nvn, where 771, . . . , rin are independently and identically distributed (i.i.d.) random signs and vi, . . . , vn are integers. The Littlewood-Offord problem

Fluctuations of extensive functions of quenched random couplings

An extensive quantity is a family of functionsΨv of random parameters, indexed by the finite regionsV (subsets of ℤd) over whichΨv are additive up to corrections satisfying the boundary estimate

Anti-concentration for Polynomials of Independent Random Variables

The results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates, and derive a general anti-concentration result on the number of copies of a fixed graph in a random graph.

A Normal Law for Matchings

This work gives asymptotic normality for any sequence of regular graphs (of positive degree) or graphs with perfect matchings, and suggests numerous related questions, some of which are discussed in the last section.

On a lemma of Littlewood and Offord

Remark. Choose Xi = l, n even. Then the interval ( — 1, + 1 ) contains Cn,m s u m s ^ i e ^ , which shows that our theorem is best possible. We clearly can assume that all the Xi are not less than 1.

Multivariate CLT follows from strong Rayleigh property

It is shown that the conclusion that the joint distribution must approach a multivariate Gaussian distribution follows already from stability of the probability generating function f, which is real stable.

Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property

  • R. Stanley
  • Mathematics
    SIAM J. Algebraic Discret. Methods
  • 1980
Techniques from algebraic geometry are used to show that certain finite partially ordered sets Q^X derived from a class of algebraic varieties X have the k-Sperner property for all k, which means that there is a simple description of the cardinality of the largest subset of $Q^X $ containing no $( k + 1 )$-element chain.