Grand Lebesgue norm estimation for binary random variables, with applications

TL;DR

This paper computes the Grand Lebesgue Space norm for binary random variables and derives sharp exponential tail bounds for sums of such variables, applicable even when they are not identically distributed.

Contribution

It introduces an exact calculation of the Grand Lebesgue Space norm for Bernoulli variables and provides new sharp tail estimates for sums of these variables under non-standard norming.

Findings

Exact Grand Lebesgue Space norm for Bernoulli variables

Sharp exponential tail bounds for sums of binary variables

Applicability to non-identically distributed variables

Abstract

We calculate the so-called Rademacher's Grand Lebesgue Space norm for a centered (shifted) indicator (Bernoulli's, binary) random variable. This norm is optimal for the centered and bounded random variables (r.v.). Using this result we derive a very simple bilateral sharp exponential tail estimates for sums of these variables, not necessary to be identical distributed, under non-standard norming, and give some examples to show the exactness of our estimates.