307 research outputs found
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
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
Bound for the maximal probability in the Littlewood-Offord problem
The paper deals with studying a connection of the Littlewood--Offord problem
with estimating the concentration functions of some symmetric infinitely
divisible distributions. It is shown that the values at zero of the
concentration functions of weighted sums of i.i.d. random variables may be
estimated by the values at zero of the concentration functions of symmetric
infinitely divisible distributions with the L\'evy spectral measures which are
multiples of the sum of delta-measures at weights involved in constructing
the weighted sums.Comment: 5 page
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