2,039 research outputs found
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
Estimates for the closeness of convolutions of probability distributions on convex polyhedra
The aim of the present work is to show that the results obtained earlier on
the approximation of distributions of sums of independent summands by the
accompanying compound Poisson laws and the estimates of the proximity of
sequential convolutions of multidimensional distributions may be transferred to
the estimation of the closeness of convolutions of probability distributions on
convex polyhedra.Comment: 8 page
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
A new bound 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.Comment: 7 pages. arXiv admin note: substantial text overlap with
arXiv:1411.687
Rare events and Poisson point processes
The aim of the present work is to show that the results obtained earlier on
the approximation of distributions of sums of independent terms by the
accompanying compound Poisson laws may be interpreted as rather sharp
quantitative estimates for the closeness between the sample containing
independent observations of rare events and the Poisson point process which is
obtained after a Poissonization of the initial sample.Comment: 9 page
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