# A Constructive Proof of a Concentration Bound for Real-Valued Random Variables

## TL;DR

This paper extends a constructive proof of a concentration bound originally for Boolean variables to real-valued variables, improving the bound and providing an efficient algorithm for identifying dependent variables.

## Contribution

It generalizes the Impagliazzo-Kabanets algorithm to real-valued variables and enhances the concentration bound by a constant factor.

## Key findings

- Extended the constructive proof to real-valued variables.
- Improved the concentration bound by a constant factor.
- Provided an efficient randomized algorithm for dependency detection.

## Abstract

Almost 10 years ago, Impagliazzo and Kabanets (2010) gave a new combinatorial proof of Chernoff's bound for sums of bounded independent random variables. Unlike previous methods, their proof is constructive. This means that it provides an efficient randomized algorithm for the following task: given a set of Boolean random variables whose sum is not concentrated around its expectation, find a subset of statistically dependent variables. However, the algorithm of Impagliazzo and Kabanets (2010) is given only for the Boolean case. On the other hand, the general proof technique works also for real-valued random variables, even though for this case, Impagliazzo and Kabanets (2010) obtain a concentration bound that is slightly suboptimal. Herein, we revisit both these issues and show that it is relatively easy to extend the Impagliazzo-Kabanets algorithm to real-valued random variables and to improve the corresponding concentration bound by a constant factor.