The role of Nitadori’s sequence in scalar quantization of general exponential sources

In 1965, Nitadori (1965) derived an infinite sequence such that for any N, the ith term of the sequence is the size of the ith half cell (counting from the right) of the N-level minimum mean-squared error quantizer for a unit variance exponential source. This was done by exploiting a key property that the source´s probability density function and its tail function are equal, a fact resulting from its memoryless property. In this paper, an asymptotic version of this key fact is found to hold for a general exponential (GE) source parameterized by an exponential power p, and it is used to show that for such a source, the size of the ith half cell of an optimal N-level quantizer multiplied by the (p-l)st power of the ith threshold approaches the ith term of the Nitadori sequence as N grows to infinity. Thus, the Nitadori sequence asymptotically characterizes the cells of MMSE quantizers for GE sources, as well as exponential.