Refining the Two-Dimensional Signed Small Ball Inequality

On the Signed Small Ball Inequality

where the implied constant is independent of n ≥ 1. The inequality above (without restriction on the coefficients) arises in connection to several areas, such as Probabilities, Approximation, and Discrepancy. With η(d) = (d − 1)/2, the inequality above follows from orthogonality, while it is conjectured that the inequality holds with η(d) = d/2. This is known and proved in (Talagrand, 1994) in ...

On the Small Ball Inequality in All Dimensions

The point of interest is the dependence upon the logarithm of the volume of the rectangles. With n(d−1)/2 on the left above, the inequality is trivial, while it is conjectured that the inequality holdswith n(d−2)/2. This is known in the case of d = 2 (Talagrand, 1994), and a recent paper of two of the authors (Bilyk and Lacey, 2006) proves a partial result towards the conjecture in three dimens...

Normic Inequality of Two-dimensional Viscosity Operator

A relation between the coefficients of Legendre expansions of two-dimensional function and those for the derived function is given. With this relation the normic inequality of two-dimensional viscosity operator is obtained.

Small Ball Probabilities for Linear Images of High Dimensional Distributions

We study concentration properties of random vectors of the form AX, where X = (X1, . . . , Xn) has independent coordinates and A is a given matrix. We show that the distribution of AX is well spread in space whenever the distributions of Xi are well spread on the line. Specifically, assume that the probability that Xi falls in any given interval of length t is at most p. Then the probability th...

The Two Dimensional, Inequality Constrained, Multi-Assignment Problem