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

Pawel Hitczenko and Robin Pemantle

Statistics & Probability Letters, 2005, vol. 72, issue 3, 249-264

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.

Keywords: Iterated; function; Random; function; Markov; chain; Coupling (search for similar items in EconPapers)
Date: 2005
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