A two-step universal VV code using an asymptotically sufficient statistic extracted from a VF code

In this paper, some theorems which give relationships between a sufficient statistic and the universality of a VV source code are presented. These theorems can be used as a methodology to construct a new two-step universal code using a sufficient statistic. As an example, the parsing tree of LZY code, which is a variation of LZ78 code, is investigated. It is proved that LZY parsing tree is an asymptotically sufficient statistic for the class of finite state sources. Furthermore, a universal two-step VV code for the same class is constructed based on the LZY parsing tree. These results are an extension of the previous results on FV codes to VV codes. Since the type (empirical distribution of symbols) is a sufficient statistic for the class of stationary memoryless sources, an asymptotically sufficient statistic can be regarded as an extension of the type, and the two-step code constructed in this paper can be seen as a generalization of Lynch-Davisson code for sources with memory.