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

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.