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

Author

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  • Hitczenko, Pawel
  • Pemantle, Robin

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.

Suggested Citation

  • Hitczenko, Pawel & Pemantle, Robin, 2005. "Central limit theorem for the size of the range of a renewal process," Statistics & Probability Letters, Elsevier, vol. 72(3), pages 249-264, May.
    • Handle: RePEc:eee:stapro:v:72:y:2005:i:3:p:249-264
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