Cantor distribution

Abstract: For a real-valued random variable whose distribution is the classical Cantor probability, the n - quantization error and the n - optimal quantization rules are calculated for every natural number n. Moreover, the connection between the rate of convergence of the logarithms of the quantization errors for n going to infinity and the Hausdorff dimension of the Cantor set is indicated.

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 $P:=\psi^2 P\circ S_1^{-1} +\psi^4 P\circ S_2^{-1}$ 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 $\psi$ 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.

Abstract: LetG0 be a (not necessarily Abelian) discrete group, and letp0 be a probability onG0. Form the direct sumG of a countable number of copies ofG0, and letp be a probability onG which is a convex combination of copies ofp0 on the factorsG0. We consider the associated random walk onG, and study the asymptotic behavior of the probabilityp*n(e) of returning to the identity elemente aftern steps. This behavior depends heavily on the choice of the convex combination, and is considerably more complicated than the behavior for finite direct sums described in, for example, Refs. 4 and 5. A by-product of the main results is a description of the asymptotic behavior of the moments of the Cantor distribution and of the moment-generating function of symmetric Bernoulli convolutions.