New Developments of Information-Spectrum Theory

信息谱法的最大意义在于它的通用性和简洁性。我们研究的目的是追求该方法的这一特点，并扩展其范围，以便为一些被认为难以构建一般理论的信息理论问题提供新的启示。主要结果如下：(1)斯普林格出版社出版了《信息论中的信息谱方法》（Information- spectrum Methods in Information Theory）的扩展译本，进一步发展了信息论的数学基础和应用的数学工具。(2)研究了一般假设检验问题的渐近性质，为经典/量子信息论提供了基础，阐明了信息谱的量子模拟。将这一结果应用于信道编码问题，说明了通用编码与充分统计量之间的密切关系。(3)从信息谱的角度分析了通用编码问题，给出了源具有通用编码的条件，阐明了通用编码与充分统计量之间的密切关系。(4)从信息谱的角度考察了信息论的“民俗”，即当任意来源的数据序列以最优速率进行无损压缩时，输出序列似乎几乎是完全随机的，并给出了明确的数学表达式及其有效性条件。

The biggest significance of the information-spectrum method lies in its generality and simplicity. The aim of our research is to pursue this characteristic of the method and to extend its scope so as to give new lights to several information-theoretical problems for which construction of general theories had been considered to be difficult. Main results are as follows.(1)An extended and translated version of the book Information-Spectrum Methods in Information Theory has been published from Springer-Verlag, which has further developed mathematical foundations of the theory as well as mathematical tools for applications.(2)We have investigated asymptotic properties of the general hypothesis testing problem, which provides a basis of classical/quantum information theory, to clarify the quantum analogue of the information spectrum. This result has been applied to the channel coding problem to elucidate a close relation between universal coding and sufficient statistics.(3)The universal coding problem has been analyzed from an information-spectrum viewpoint, which has given a condition for a source to have a universal code and elucidated a close relation between universal coding and sufficient statistics.(4)We have investigated from an information-spectrum viewpoint the "folklore" of information theory that the output sequence seems to be almost completely random when a data sequence from an arbitrary source is losslessly compressed with the optimal rate, whereby the folklore has been given a clear mathematical expression and a condition for its validity.