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Approximation and invariance properties of one-dimensional probabilities
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- Author / Creator
- Xu,Chuang
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Approximation of probability measures, quantization, Kantorovich metric, Levy metric, Kolmogorov metric, Benford's Law, slowly changing sequences, asymptotic distribution, invariance property.","This thesis is based on four papers. The first two papers fall into the field of approximation of one-dimensional probabilities, the third into uniform distribution theory, and the last paper relates to ergodic theory.
The first paper (joint with Arno Berger), \hfill studies best finitely supported approximations\ \noindent of one-dimensional probability measures with respect to (w.r.t.) the $L^r$-Kantorovich (or transport) distance, where either the
locations or the weights of the approximations' atoms are
prescribed. Special attention is given to the case of
best uniform approximations (i.e., all atoms having equal weight).In the second paper (joint with Arno Berger), for arbitrary one-dimensional Borel probability measures with compact
support, characterizations are established of the best finitely
supported approximations, relative to three familiar probability
metrics (L\'{e}vy, Kantorovich, and Kolmogorov), given any number of
atoms, and allowing for additional constraints regarding weights or
positions of atoms. As an application, best (constrained or
unconstrained) approximations are identified for Benford's Law
(logarithmic distribution of significands).The third paper studies the distributional asymptotics of the slowly changing sequence of logarithms $(\log_bn)$ with integer base $b\ge2.$ An upper estimate $\lt(N^{-1}\lt(\log N\rt)^{1/2}\rt)$ is obtained for the rate of convergence w.r.t. the Kantorovich metric on the circle. Moreover, a sharp rate of convergence $\lt(N^{-1}\log N\rt)$ w.r.t. the Kantorovich and the discrepancy (or Kolmogorov) metrics on the real line is derived.
The last paper proves a threshold result on the existence of a circularly invariant and uniform probability measure (CIUPM) for non-constant linear transformations on the real line, which shows that there is a constant $c$ depending only on the slope of the linear transformation such that there exists a CIUPM if and only if the diameter of the support is not smaller than $c$. Moreover, the CIUPM is unique up to translation when the diameter of the support is equal to $c.$
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- Graduation date
- Fall 2018
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- Type of Item
- Thesis
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- Degree
- Doctor of Philosophy
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- License
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