### Abstract

The second-order achievable asymptotics in typical random number generation problems such as resolvability, intrinsic randomness, and fixed-length source coding are considered. In these problems, several researchers have derived the first-order and the second-order achievability rates for general sources using the information spectrum methods. Although these formulas are general, their computations are quite hard. Hence, an attempt to address explicit computation problems of achievable rates is meaningful. In particular, for i.i.d. sources, the second-order achievable rates have earlier been determined simply by using the asymptotic normality. In this paper, we consider mixed sources of two i.i.d. sources. The mixed source is a typical case of nonergodic sources and whose self-information does not have the asymptotic normality. Nonetheless, we can explicitly compute the second-order achievable rates for these sources on the basis of two-peak asymptotic normality. In addition, extensions of our results to more general mixed sources, such as a mixture of countably infinite i.i.d. sources or Markov sources, and a continuous mixture of i.i.d. sources, are considered.

Original language | English |
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Article number | 6289366 |

Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | IEEE Transactions on Information Theory |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Jan 7 |

Externally published | Yes |

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### Keywords

- Asymptotic normality
- mixed source
- random number generation
- second-order achievability
- source coding

### ASJC Scopus subject areas

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

### Cite this

*IEEE Transactions on Information Theory*,

*59*(1), 1-16. [6289366]. https://doi.org/10.1109/TIT.2012.2215836