Optimal quantization for the Cantor distribution generated by infinite similutudes

Abstract

Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {S_j : 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 {S_j : 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.