### Abstract

The entropy power inequality (EPI) yields lower bounds on the differential entropy of the sum of two independent real-valued random variables in terms of the individual entropies. Versions of the EPI for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy, but no universal inequality is known (beyond trivial ones). More recently, the sumset theory for the entropy function yields a sharp inequality H(X+X′)-H(X)≥1/2-o(1) when X + X′ are independent identically distributed (i.i.d.) with high entropy. This paper provides the inequality H(X+X′)-H(X)≥ g(H(X)) , where X + X′ are arbitrary i.i.d. integer-valued random variables and where g is a universal strictly positive function on ℝ_{+} satisfying g(0)=0. Extensions to nonidentically distributed random variables and to conditional entropies are also obtained.

Original language | English (US) |
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Article number | 6797921 |

Pages (from-to) | 3787-3796 |

Number of pages | 10 |

Journal | IEEE Transactions on Information Theory |

Volume | 60 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2014 |

### All Science Journal Classification (ASJC) codes

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

### Keywords

- Entropy inequalities
- Mrs. Gerber's Lemma
- Shannon sumset theory
- doubling constant
- entropy power inequality

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## Cite this

*IEEE Transactions on Information Theory*,

*60*(7), 3787-3796. [6797921]. https://doi.org/10.1109/TIT.2014.2317181