Quantization for a set of discrete distributions on the set of natural numbers

Abstract: The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. In this paper, first we state and prove a theorem, and then give a conjecture. We verify the conjecture by a few examples. Assuming that the conjecture is true, for a set of discrete distributions on the set of natural numbers we have calculated the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. In addition, the quantization dimension is also calculated.