Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions
Mrinal Kanti Roychowdhury
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4555-4570 doi: 10.3934/dcds.2018199

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. Let $P$ be a Borel probability measure on $\mathbb R$ such that $P = \frac 12 P\circ S_1^{-1}+\frac 12 P\circ S_2^{-1},$ where $S_1$ and $S_2$ are two contractive similarity mappings given by $S_1(x) = rx$ and $S_2(x) = rx+1-r$ for $0<r<\frac 12$ and $x∈ \mathbb R$. Then, $P$ is supported on the Cantor set generated by $S_1$ and $S_2$. The case $r = \frac 13$ was treated by Graf and Luschgy who gave an exact formula for the unique optimal quantization of the Cantor distribution $P$ (Math. Nachr., 183 (1997), 113-133). In this paper, we compute the precise range of $r$-values to which Graf-Luschgy formula extends.

keywords: Cantor set probability distribution optimal quantizers quantization error
Quantization coefficients for ergodic measures on infinite symbolic space
Mrinal Kanti Roychowdhury
Discrete & Continuous Dynamical Systems - A 2014, 34(7): 2829-2846 doi: 10.3934/dcds.2014.34.2829
In this paper we consider an ergodic measure with bounded distortion on a symbolic space generated by an infinite alphabet, and showed that for each $r\in (0, +\infty)$ there exists a unique $k_r \in (0, +\infty)$ such that both the $k_r$-dimensional lower and upper quantization coefficients for its image measure $m$ with the support lying on the limit set generated by an infinite conformal iterated function system satisfying the strong open set condition are finite and positive. In addition, it shows that $k_r$ can be expressed by a simple formula involving the temperature function of the system. The result extends and generalizes a similar result of Roychowdhury established for a finite conformal iterated function system [Bull. Polish Acad. Sci. Math. 57 (2009)].
keywords: Quantization coefficients temperature function. ergodic measures with bounded distortion quantization dimension topological pressure
Nearly continuous Kakutani equivalence of adding machines
Mrinal Kanti Roychowdhury Daniel J. Rudolph
Journal of Modern Dynamics 2009, 3(1): 103-119 doi: 10.3934/jmd.2009.3.103
One says that two ergodic systems $(X,\mathcal F,\mu)$ and $(Y,\mathcal G,\nu)$ preserving a probability measure are evenly Kakutani equivalent if there exists an orbit equivalence $\phi: X\to Y$ such that, restricted to some subset $A\subseteq X$ of positive measure, $\phi$ becomes a conjugacy between the two induced maps $T_A$ and $S_{\phi(A)}$. It follows from the general theory of loosely Bernoulli systems developed in [8] that all adding machines are evenly Kakutani equivalent, as they are rank-1 systems. Recent work has shown that, in systems that are both topological and measure-preserving, it is natural to seek to strengthen purely measurable results to be "nearly continuous''. In the case of even Kakutani equivalence, what one asks is that the map $\phi$ and its inverse should be continuous on $G_\delta$ subsets of full measure and that the set $A$ should be within measure zero of being open and of being closed. What we will show here is that any two adding machines are indeed equivalent in this nearly continuous sense.
keywords: Nearly continuous Kakutani equivalence adding machine.

Year of publication

Related Authors

Related Keywords

[Back to Top]