A well-known longstanding conjecture on the supremum of the tails of normalized sums of independent Rademacher random variables is disproved. A special case of this conjecture was recently disproved by A. Zhubr.

An exact Rosenthal-type inequality for the third absolute moments is given, as well as a number of related results. Such results are useful in applications to Berry–Esseen bounds.

In the note we study large and superlarge deviation probabilities of sum of i.i.d. lattice random variables, whose distribution function has an exponentially decreasing tail at infinity.