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

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 fn∘⋯∘f1 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.