# Quantization Dimension for Probability Definitions

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### Description

The term quantization refers to the process of estimating a given probability by a discrete probability supported on a finite set. The quantization dimension Dr of a probability is related to the asymptotic rate at which the expected distance (raised to the rth power) to the support of the quantized version of the probability goes to zero as the size of the support is allowed to go to infinity. This assumes that the quantized versions are in some sense ``optimal'' in that the expected distances have been minimized. In this dissertation we give a short history of quantization as well ... continued below

### Creation Information

Lindsay, Larry J. December 2001.

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• Lindsay, Larry J.

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### Description

The term quantization refers to the process of estimating a given probability by a discrete probability supported on a finite set. The quantization dimension Dr of a probability is related to the asymptotic rate at which the expected distance (raised to the rth power) to the support of the quantized version of the probability goes to zero as the size of the support is allowed to go to infinity. This assumes that the quantized versions are in some sense ``optimal'' in that the expected distances have been minimized. In this dissertation we give a short history of quantization as well as some basic facts. We develop a generalized framework for the quantization dimension which extends the current theory to include a wider range of probability measures. This framework uses the theory of thermodynamic formalism and the multifractal spectrum. It is shown that at least in certain cases the quantization dimension function D(r)=Dr is a transform of the temperature function b(q), which is already known to be the Legendre transform of the multifractal spectrum f(a). Hence, these ideas are all closely related and it would be expected that progress in one area could lead to new results in another. It would also be expected that the results in this dissertation would extend to all probabilities for which a quantization dimension function exists. The cases considered here include probabilities generated by conformal iterated function systems (and include self-similar probabilities) and also probabilities generated by graph directed systems, which further generalize the idea of an iterated function system.

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### Creation Date

• December 2001

### Added to The UNT Digital Library

• Sept. 25, 2007, 10:40 p.m.

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• Jan. 14, 2014, 4:08 p.m.

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Lindsay, Larry J. Quantization Dimension for Probability Definitions, dissertation, December 2001; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc3008/: accessed June 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .