Analytic Variations on Redundancy Rates of Renewal Processes

Projet ALGORITHMESCsiszár and Shields have recently proved that the minimax redundancy for a class of renewal processes is $\Theta(\sqrt{n})$ where $n$ is the block length. This interesting result provides a first non-trivial bound on redundancy for a non-parametric family of processes. The present paper provides a precise estimate up to the constant term of the redundancy rate for such sources. The asymptotic expansion is derived by complex--analytic methods that include generating function representations, Mellin ransforms, singularity analysis and saddle point estimates. This work places itself within the framework of analytic information theory

Universal Coding on Infinite Alphabets: Exponentially Decreasing Envelopes

This paper deals with the problem of universal lossless coding on a countable infinite alphabet. It focuses on some classes of sources defined by an envelope condition on the marginal distribution, namely exponentially decreasing envelope classes with exponent $\alpha$. The minimax redundancy of exponentially decreasing envelope classes is proved to be equivalent to $\frac{1}{4 \alpha \log e} \log^2 n$. Then a coding strategy is proposed, with a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive algorithm is provided, whose redundancy is equivalent to the minimax redundanc

Analytic variations on redundancy rates of renewal processes

Theme 2 - Genie logiciel et calcul symbolique. Projet AlgorithmesSIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1998 n.3553 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Analytic Variations on Redundancy Rates of Renewal Processes

Csiszár and Shields have recently proved that the minimax redundancy for a class of renewal processes is \Theta( p n) where n is the block length. This interesting result provides a first non-trivial bound on redundancy for a non-parametric family of processes. The present paper provides a precise estimate up to the constant term of the redundancy rate for such sources. The asymptotic expansion is derived by complex--analytic methods that include generating function representations, Mellin transforms, singularity analysis and saddle point estimates. This work places itself within the framework of analytic information theory

Analytic Variations on Redundancy Rates of Renewal Processes

Csisz'ar and Shields have recently proved that the minimax redundancy for a class of renewal processes is \Theta( p n) where n is the block length. This interesting result provides a first non-trivial bound on redundancy for a nonparametric family of processes. The present paper provides a precise estimate of the redundancy rate for such sources, namely, 2 log 2 r i ß 2 6 \Gamma 1 j n \Gamma 5 8 log 2 n + 1 2 log 2 log n +O(1): This asymptotic expansion is derived by complex--analytic methods that include generating function representations, Mellin transforms, singularity analysis and saddle point estimates. This work places itself within the framework of analytic information theory. Keywords--- Redundancy, universal coding, renewal processes, partitions of integers, tree function, Mellin transform, saddle point method, analytic information theory. I. Introduction R ECENT YEARS have seen a resurgence of interest in redundancy rates of lossless coding; see [3], [13], [15]..