Maximal probability inequalities for weighted sums of Rademachers.

Язык доклада – русский
In this talk we consider the sums
$$ S_n=a_1\varepsilon_1+a_1\varepsilon_1+\dots+a_n\varepsilon_n $$
of weighted Rademacher random variables $\varepsilon_1,\dots,\varepsilon_n$ such that
$$ \mathbb P[\varepsilon_i=1]=\mathbb P[\varepsilon_i=-1]=\frac12. $$

In the first part of the talk, the tail probabilities under condition that the variance of the sum $S_n$ is bounded by 1 are considered. In the second part, a non-uniform Littlewood-Offord inequality for $S_n$ in the case when $a_i$ are bounded will be presented (joint work with Tomas Jus̆kevic̆ius).