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| HAL: hal-00121892, version 2 |
| arXiv: 0801.2456 |
| Detailed view |  Export this paper |
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| Available versions: | v1 (2006-12-22) | v2 (2008-01-16) |
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| Coding on countably infinite alphabets |
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| Stéphane Boucheron 1Aurélien Garivier 2 |
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| (2006) |
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| This paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets. Classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks for coding techniques originating from the theory of universal coding over finite alphabets. We prove general upper-bounds on minimax regret and lower-bounds on minimax redundancy for such source classes. The general upper bounds emphasize the role of the Normalized Maximum Likelihood codes with respect to minimax regret in the infinite alphabet context. Lower bounds are derived by tailoring sharp bounds on the redundancy of Krichevsky-Trofimov coders for sources over finite alphabets. Up to logarithmic (resp. constant) factors the bounds are matching for source classes defined by algebraically declining (resp. exponentially vanishing) envelopes. Effective and (almost) adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing envelopes. Those results extend ourknowledge concerning universal coding to contexts where the key tools from parametric inference |
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| 1: | Laboratoire de Probabilités et Modèles Aléatoires (LPMA) |
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CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot |
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| 2: | Laboratoire de Mathématiques d'Orsay (LM-Orsay) |
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CNRS : UMR8628 – Université Paris XI - Paris Sud |
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| Subject | : | Mathematics/Statistics Statistics/Statistics Theory |
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| nml – countable alphabets – redundancy – adaptive compression |
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| hal-00121892, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00121892 | |
| oai:hal.archives-ouvertes.fr:hal-00121892 | |
| From: Aurélien Garivier | |
| Submitted on: Wednesday, 16 January 2008 09:41:45 | |
| Updated on: Wednesday, 16 January 2008 09:52:34 | |
