Optimal quantization for nonuniform Cantor distributions

Abstract

Let $P$ be a Borel probability measure on $\mathbb R$ such that $P=\frac 1 4 P\circ S_1^{-1} +\frac 3 4 P\circ S_2^{-1}$, where $S_1$ and $S_2$ are two similarity mappings on $\mathbb R$ such that $S_1(x)=\frac 1 4 x $ and $S_2(x)=\frac 1 2 x +\frac 12$ for all $x\in \mathbb R$. Such a probability measure $P$ has support the Cantor set generated by $S_1$ and $S_2$. For this probability measure, in this paper, we give an induction formula to determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. We have shown that the same induction formula also works for the Cantor distribution $P:=ψ^2 P\circ S_1^{-1} +ψ^4 P\circ S_2^{-1}$ supported by the Cantor set generated by $S_1(x)=\frac 13x$ and $S_2(x)=\frac 13 x+\frac 23$ for all $x\in \mathbb R$, where $ψ$ is the square root of the Golden ratio $\frac 12(\sqrt 5-1)$. In addition, we give a counter example to show that the induction formula does not work for all Cantor distributions. Using the induction formula we obtain some results and observations which are also given in this paper.