Coupling results and Markovian structures for number representations of continuous random variables

A general setting for nested subdivisions of a bounded real set into intervals defining the digits X₁,X₂,... of a random variable X with a probability density function f is considered. Under the weak condition that f is almost everywhere lower semi-continuous, a coupling between X and a non-negative integer-valued random variable N is established so that X₁,...,XN have an interpretation as the ``sufficient digits'', since the distribution of R=(XN+1,XN+2,...) conditioned on S=(X₁,...,XN) does not depend on f. Adding a condition about a Markovian structure of the lengths of the intervals in the nested subdivisions, $R\,|\,S$ becomes a Markov chain of a certain order s≥0. If $s=0$ then XN+1,XN+2,... are IID with a known distribution. When $s>0$ and the Markov chain is uniformly geometric ergodic, a coupling is established between $(X,N)$ and a random time M so that the chain after time max,s\+M-s is stationary and M follows a simple known distribution. The results are related to several examples of number representations generated by a dynamical system, including base-q expansions, generalized L\"uroth series, β-expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.