173 research outputs found
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Non-abelian Littlewood-Offord inequalities
In 1943, Littlewood and Offord proved the first anti-concentration result for
sums of independent random variables. Their result has since then been
strengthened and generalized by generations of researchers, with applications
in several areas of mathematics.
In this paper, we present the first non-abelian analogue of Littlewood-Offord
result, a sharp anti-concentration inequality for products of independent
random variables.Comment: 14 pages Second version. Dependence of the upper bound on the matrix
size in the main results has been remove
Optimal Inverse Littlewood-Offord theorems
Let eta_i be iid Bernoulli random variables, taking values -1,1 with
probability 1/2. Given a multiset V of n integers v_1,..., v_n, we define the
concentration probability as rho(V) := sup_{x} Pr(v_1 eta_1+...+ v_n eta_n=x).
A classical result of Littlewood-Offord and Erdos from the 1940s asserts that
if the v_i are non-zero, then rho(V) is O(n^{-1/2}). Since then, many
researchers obtained improved bounds by assuming various extra restrictions on
V. About 5 years ago, motivated by problems concerning random matrices, Tao and
Vu introduced the Inverse Littlewood-Offord problem. In the inverse problem,
one would like to give a characterization of the set V, given that rho(V) is
relatively large. In this paper, we introduce a new method to attack the
inverse problem. As an application, we strengthen a previous result of Tao and
Vu, obtaining an optimal characterization for V. This immediately implies
several classical theorems, such as those of Sarkozy-Szemeredi and Halasz. The
method also applies in the continuous setting and leads to a simple proof for
the beta-net theorem of Tao and Vu, which plays a key role in their recent
studies of random matrices. All results extend to the general case when V is a
subset of an abelian torsion-free group and eta_i are independent variables
satisfying some weak conditions
Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem: a shortened version
Let be independent identically distributed random
variables. In this paper we study the behavior of concentration functions of
weighted sums with respect to the arithmetic structure
of coefficients~ in the context of the Littlewood--Offord problem.
Concentration results of this type received renewed interest in connection with
distributions of singular values of random matrices. Recently, Tao and Vu
proposed an Inverse Principle in the Littlewood--Offord problem. We discuss the
relations between the Inverse Principle of Tao and Vu as well as that of Nguyen
and Vu and a similar principle formulated for sums of arbitrary independent
random variables in the work of Arak from the 1980's. This paper is a shortened
and edited version of the preprint arXiv:1506.09034. Here we present the
results without proofs.Comment: 9 pages. shortened version of arXiv:1506.0903
Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices
Let denote a random symmetric by matrix, whose upper diagonal
entries are iid Bernoulli random variables (which take value -1 and 1 with
probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show
that is non-singular with probability for any positive
constant . The proof uses an inverse Littlewood-Offord result for quadratic
forms, which is of interest of its own.Comment: Some minor corrections in Section 10 of v
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