Quantization for a Condensation System

Shivam Dubey, Mrinal Kanti Roychowdhury () and Saurabh Verma
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Shivam Dubey: Department of Applied Sciences, Indian Institute of Information Technology Allahabad, Prayagraj 211015, UP, India
Mrinal Kanti Roychowdhury: School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA
Saurabh Verma: Department of Applied Sciences, Indian Institute of Information Technology Allahabad, Prayagraj 211015, UP, India

Mathematics, 2025, vol. 13, issue 9, 1-43

Abstract: For a given r ∈ ( 0 , + ∞ ) , the quantization dimension of order r , if it exists, denoted by D r ( μ ) , represents the rate at which the n th quantization error of order r approaches zero as the number of elements n in an optimal set of n -means for μ tends to infinity. If D r ( μ ) does not exist, we define D ̲ r ( μ ) and D ¯ r ( μ ) as the lower and the upper quantization dimensions of μ of order r , respectively. In this paper, we investigate the quantization dimension of the condensation measure μ associated with a condensation system ( { S j } j = 1 N , ( p j ) j = 0 N , ν ) . We provide two examples: one where ν is an infinite discrete distribution on R , and one where ν is a uniform distribution on R . For both the discrete and uniform distributions ν , we determine the optimal sets of n -means, calculate the quantization dimensions of condensation measures μ , and show that the D r ( μ ) -dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive.

Keywords: condensation measure; optimal quantizers; quantization error; quantization dimension; quantization coefficient; discrete distribution; uniform distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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