• 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 fX . We show that if all of the zeros ζ of fX satisfy | arg(ζ)| > δ and R 6 |ζ| 6 R then Var(X) > cRn, where c > 0 is a absolute constant. We show that this result is sharp, up to the factor 2 in the exponent of R. As a… 



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

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

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.


We derive an asymptotic expansion as n →∞ for a large range of coefficients of (f (z)) n ,w here f( z) is a power series satisfying | f( z)| <f (|z|) for z ∈ C, z ∉R + .W henf is a polynomial and the

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.

A characterization of polynomials whose high powers have non-negative coefficients

Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In

A general method for lower bounds on fluctuations of random variables

A general method for lower bounds on fluctuations of random variables is introduced and is used to obtain new results for the stochastic traveling salesman problem, the Stochastic minimal matching problems, the random assignment problem, and the Sherrington-Kirkpatrick model of spin glasses, first-passage percolation and random matrices.