## Abstract

The asymptotic a.s.-relation H = lim_{n→∞} n log n ÷ Σ_{i=1}^{n} L_{i}^{n} (X) is derived for any finite-valued stationary ergodic process X = (X_{n}, n ∈ Z) that satisfies a Doeblin-type condition: there exists r ≥ 1 such that ess_{x}inf P(X_{n+r} | x_{-∞,n}) ≥ α > 0. Here, H is the entropy rate of the process X, and L_{i}^{n}(X) is the length of a shortest prefix in X which is initiated at time i and is not repeated among the prefixes initiated at times j, 1 ≤ i ≠ j ≤ n. The validity of this limiting result was established by Shields in 1989 for i.i.d. processes and also for irreducible aperiodic Markov chains. Under our new condition, we prove that this holds for a wider class of processes, that may have infinite memory.

Original language | English (US) |
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State | Published - 1994 |

Event | Proceedings of the 1994 IEEE International Symposium on Information Theory - Trodheim, Norw Duration: Jun 27 1994 → Jul 1 1994 |

### Other

Other | Proceedings of the 1994 IEEE International Symposium on Information Theory |
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City | Trodheim, Norw |

Period | 6/27/94 → 7/1/94 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Information Systems
- Modeling and Simulation
- Applied Mathematics