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Optimal quantization for the Cantor distribution generated by infinite similutudes

  • Mrinal Kanti RoychowdhuryEmail author
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Abstract

Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that \(P=\Sigma_{j=1}^{\infty}\frac{1}{2^{j}}P\circ{S}_j^{-1}\), where for each j ∈ ℕ and x ∈ ℝ, \(S_j(x)=\frac{1}{3^j}x+1-\frac{1}{3^{j-1}}\). Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector \((\frac{1}{2},\frac{1}{{{2^2}}},...)\), for all n ∈ ℕ, we determine the optimal sets of n-means and the nth quantization errors for the infinite self-similar measure P. The technique obtained in this paper can be utilized to determine the optimal sets of n-means and the nth quantization errors for more general infinite self-similar measures.

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesUniversity of Texas Rio Grande ValleyEdinburgUSA

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