# Sharp Bounds for Mutual Covering

## TL;DR

This paper introduces new sharp bounds for the mutual covering lemma using advanced concentration inequalities, with applications showing rapid convergence and exact error exponents in distribution simulation.

## Contribution

It develops novel mutual covering bounds employing Talagrand's inequality and applies them to derive exponential convergence rates and precise error exponents.

## Key findings

- Covering probability converges doubly exponentially fast when set probability is bounded away from 0 and 1.
- The maximum difference in probabilities relates to the total variation distance between distributions.
- Exact error exponents are derived for the joint distribution simulation problem.

## Abstract

A fundamental tool in network information theory is the covering lemma, which lower bounds the probability that there exists a pair of random variables, among a give number of independently generated candidates, falling within a given set. We use a weighted sum trick and Talagrand's concentration inequality to prove new mutual covering bounds. We identify two interesting applications: 1) When the probability of the set under the given joint distribution is bounded away from 0 and 1, the covering probability converges to 1 \emph{doubly} exponentially fast in the blocklength, which implies that the covering lemma does not induce penalties on the error exponents in the applications to coding theorems. 2) Using Hall's marriage lemma, we show that the maximum difference between the probability of the set under the joint distribution and the covering probability equals half the minimum total variation distance between the joint distribution and any distribution that can be simulated by selecting a pair from the candidates. Thus we use the mutual covering bound to derive the exact error exponent in the joint distribution simulation problem. In both applications, the determination of the exact exponential (or doubly exponential) behavior relies crucially on the sharp concentration inequality used in the proof of the mutual covering lemma.