The Entropy of Recursively-Indexed Geometric Distribution

This paper proves by straightforward computation an interesting property of a recursive indexing: it preserves the entropy of a geometrically-distributes stationary memoryless source. This result is a pleasant surprise because the recursive indexing though one-to-one, is a symbol-to-string mapping and the entropy is measured in terms of the source symbols. This preservation of the entropy implies that the minimum average number of bits needed to represent a geometric memoryless source by the recursive indexing followed by a good binary encoder of a finite imput alphabet remains the same as that by a good encoder of an infinite input alphabet. Therefore, the recursive indexing theoretically keeps coding optimality intact. For this reason recursive indexing can provide an interface for a binary code with a finite code book that performs reasonably well for a source with an infinite alphabet.