# Seminar in applied mathematics and statistics

SPEAKER:  Jesper Møller (University of Aalborg).

TITLE: Characterization and construction of singular distribution functions for random base-q expansions with the digits forming a stationary process.

ABSTRACT:  Consider the cumulative distribution function (CDF) $F$ of a random variable given by the base-$q$ expansion $\sum_{n=1}^\infty X_n q^{-n}$, where $q\ge2$ is an integer and each random variable $X_n\in \{0,\dots,q-1\}$. We show that stationarity of the stochastic process $\{X_n\}_{n\geq 1}$ is equivalent to a certain functional equation obeyed by $F$, which enables us to give a complete characterization of the structure of $F$. In particular, we prove that the absolutely continuous component of $F$ can only be the uniform distribution on the unit interval while its discrete component can only be a countable convex combination of certain explicitly computable CDFs for distributions with finite support. Moreover, we show that for a large class of stationary stochastic processes, their corresponding $F$ is singular (that is, $F'=0$ almost everywhere) and continuous; and often also strictly increasing. We also consider geometric constructions and `relatively closed form expressions' of $F$. Finally, we study special cases of models, including stationary Markov chains of any order, stationary renewal point processes, and mixtures of such models, where expressions and plots of $F$ will be exemplified.

Joint work with Horia Cornean, Ira W. Herbst, Benjamin Støttrup, and Kasper S. Sørensen.

----

Upcoming events (after November 27):

Wednesday, January 15 at 15.15: David Meder and Ollie Hulme

Wednesday, February 19 at 15.15: Budhi Surya