Quantization for a condensation system

2025/03/18 by Dubey, Shivam, Roychowdhury, Mrinal Kanti, Verma, Saurabh
#28A80 #60Exx #94A34 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.14344

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 nth quantization error of order r approaches to 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 \underlineD_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, and 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.

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