Sharp bounds for mutual covering

Jingbo Liu, Mohammad Hossein Yassaee, Sergio Verdu

Research output: Contribution to journalArticlepeer-review


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 given 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 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; and 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 double exponential) behavior relies crucially on the sharp concentration inequality used in the proof of the mutual covering lemma.

Original languageEnglish (US)
Article number8725594
Pages (from-to)8067-8083
Number of pages17
JournalIEEE Transactions on Information Theory
Issue number12
StatePublished - Dec 2019
Externally publishedYes


  • concentration inequalities
  • covering lemmas
  • distribution simulation
  • information density
  • network information theory
  • one-shot method
  • randomness generation
  • rejection sampling
  • Shannon theory

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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