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Central limit theorem for the size of the range of a renewal process

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  • Hitczenko, Pawel
  • Pemantle, Robin
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    Abstract

    We study the range of a Markov chain moving forward on the positive integers. For every position, there is a probability distribution on the size of the next forward jump. Taking a scaling limit as the means and variances of these distributions approach given continuous functions of position, there is a Gaussian limit law for the number of sites hit in a given rescaled interval. We then apply this to random coupling. At each time, n, a random function fn is applied to the set {1,...,N}. The range Rn of the composition fno...of1 shrinks as n increases. A Gaussian limit law for the total number of values of Rn follows from the limit law together with an extension to non-compact rescaled ranges.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 72 (2005)
    Issue (Month): 3 (May)
    Pages: 249-264

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    Handle: RePEc:eee:stapro:v:72:y:2005:i:3:p:249-264

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    Keywords: Iterated function Random function Markov chain Coupling;

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