Order statistics for the Cantor-Fibonacci distribution

Summary.

The Cantor distribution is a probability distribution whose cumulative distribution function is the Cantor function. It is obtained from strings consisting of letters 0 and 1 and appropriately attaching a value to them. The Cantor–Fibonacci distribution additionally rejects strings with two adjacent letters 1. A probability model is associated by assuming that each admissible string (word) of length m is equally likely; eventually the limit m → ∞ is considered. In this way, one can work with discrete objects, which might not be strictly necessary, but is easy to understand.

We assume that n random numbers (values of random strings) are drawn independently. The interest is in order statistics of these n values: the (average of) the smallest resp. largest of them. Recursions are obtained which are evaluated asymptotically.

Generalisations to the d-smallest resp. d-largest element are also considered.