On some distribution related to digital expansions

Ligia-Loretta Cristea

Johannes Kepler Universität Linz, Austria

Coauthor(s): Helmut Prodinger

All results presented here stem from joint work with Helmut Prodinger.
First, we study certain probability measures of binomial type defined in a recursive way on the unit interval. These measures are related to the sum-of-digit function and similar quantities. In particular, we undertake an asymptotic analysis of the moments of the corresponding distributions.
Subsequently, we consider certain aspects of the Cantor-Fibonacci distribution. 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 $m\to\infty$ is considered.

We assume that $n$ random numbers (values of random strings) are drawn independently. We study order statistics of these $n$ values: the (average of) the smallest resp. largest of them.

Supported by the Austrian Science Foundation (FWF), project S9609, ``Analytic and Probabilistic Methods in Combinatorics''.