4,099 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
Experimental evidence of ageing and slow restoration of the weak-contact configuration in tilted 3D granular packings
Granular packings slowly driven towards their instability threshold are
studied using a digital imaging technique as well as a nonlinear acoustic
method. The former method allows us to study grain rearrangements on the
surface during the tilting and the latter enables to selectively probe the
modifications of the weak-contact fraction in the material bulk. Gradual ageing
of both the surface activity and the weak-contact reconfigurations is observed
as a result of repeated tilt cycles up to a given angle smaller than the angle
of avalanche. For an aged configuration reached after several consecutive tilt
cycles, abrupt resumption of the on-surface activity and of the weak-contact
rearrangements occurs when the packing is subsequently inclined beyond the
previous maximal tilting angle. This behavior is compared with literature
results from numerical simulations of inclined 2D packings. It is also found
that the aged weak-contact configurations exhibit spontaneous restoration
towards the initial state if the packing remains at rest for tens of minutes.
When the packing is titled forth and back between zero and near-critical
angles, instead of ageing, the weak-contact configuration exhibits "internal
weak-contact avalanches" in the vicinity of both the near-critical and zero
angles. By contrast, the stronger-contact skeleton remains stable
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
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