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    Clumping, Stick-breaking, and an Inhomogeneous Markov Chain

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    Author
    Lippitt, William Lindsay
    Issue Date
    2019
    Keywords
    clumping
    Dirichlet
    GEM
    Markov
    RAM
    stick-breaking
    Advisor
    Sethuraman, Sunder
    
    Metadata
    Show full item record
    Publisher
    The University of Arizona.
    Rights
    Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction, presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
    Abstract
    We study the connections among three types of objects: Residual Allocation Models (RAMs), a generalized class of stick-breaking processes which include Dirichlet processes, and the occupation laws of certain discrete space time-inhomogeneous Markov chains. These connections are established through a new clumping procedure on RAMs according to a clumping sequence. We introduce and analyze two methods of its application with respect to homogeneous Markov chains. Clumping according to the rst method results in a class of stick-breaking measures we later identify as the limit of the empirical occupation measures for particular time-inhomogeneous Markov chains. Clumping according to the second method leads to a general study of self-similarity as a characterization of random probability measures, which we apply in context. We further discuss the occupation law of an inhomogeneous Markov chain related to simulated annealing and its connections to stick-breaking measures. By introducing a reverse-chronological clumping procedure, we dene and study the local occupations up to time n of the Markov chain. As n gets large, these local occupations converge jointly to the clumped RAM structure resulting from the rst method above. We then explore additional properties of the generalized stick-breaking measure uncovered by the connection to the occupation law, and consider potential applications of results.
    Type
    text
    Electronic Dissertation
    Degree Name
    Ph.D.
    Degree Level
    doctoral
    Degree Program
    Graduate College
    Mathematics
    Degree Grantor
    University of Arizona
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