Constrained quantization for discrete distributions (2507.07923v1)

Abstract: Constrained quantization for a Borel probability measure refers to the process of approximating a given probability distribution by a discrete probability measure supported on a finite number of points that lie within a specified set, known as the constraint. This paper investigates constrained quantization for several types of discrete probability distributions under two different constraints. The study begins with an analysis of two discrete distributions, one uniform and one nonuniform, defined over a finite support. For each distribution, the constrained optimal sets of representative points and the corresponding constrained quantization errors are computed under both constraints. The analysis is then extended to an infinite discrete distribution supported on the reciprocals of the natural numbers. For this infinite case, and under two different constraints, the paper determines all possible constrained optimal sets of $n$-points and the associated $n$th constrained quantization errors for all $1 \leq n \leq 2000$. The problem of constrained quantization for $n > 2000$ in this infinite setting remains open.