### Abstract

Renyi's entropy and divergence of order α are given operational characterizations in terms of block coding and hypothesis testing, as so-called β-cutoff rates, with α = 1/1 + β for entropy and α = 1/1 - β for divergence. Out of several possible definitions of mutual information of order α (for channel W and input distribution P) we adopt I_{α}(P,W) = min_{Q}Σ_{x}P(x)D_{α}(W(·|x)∥Q). This admits interpretation as a β-cutoff rate, with α = 1/1 - β (at least for α ≥ 1/2 ), and so does max_{p}I_{α}(P,W), the 'Renyi capacity.' Geometrically, the β-cutoff rate for a discrete memoryless source or channel is the τ-axis intercept of the tangent of slope β to the curve e(r), where c(τ) is the exponent of the probability of error resp, of correct decoding for the best codes of rate r, according as r is an achievable rate or not. The ordinary cutoff rate of a DMC is the β-cutoff rate with β = -1. The β-cutoff rate for hypothesis testing has a similar geometric representation, c(τ) being the exponent of convergence of the probability of type 2 error to 0 or 1, for the best tests of sample size n → ∞ with probability exp(-nτ) of type 1 error.

Original language | English |
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Title of host publication | Proceedings of the 1993 IEEE International Symposium on Information Theory |

Publisher | Publ by IEEE |

Number of pages | 1 |

ISBN (Print) | 0780308786 |

Publication status | Published - Jan 1 1993 |

Event | Proceedings of the 1993 IEEE International Symposium on Information Theory - San Antonio, TX, USA Duration: Jan 17 1993 → Jan 22 1993 |

### Publication series

Name | Proceedings of the 1993 IEEE International Symposium on Information Theory |
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### Other

Other | Proceedings of the 1993 IEEE International Symposium on Information Theory |
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City | San Antonio, TX, USA |

Period | 1/17/93 → 1/22/93 |

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### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proceedings of the 1993 IEEE International Symposium on Information Theory*(Proceedings of the 1993 IEEE International Symposium on Information Theory). Publ by IEEE.