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 $X_1,X_2,...$ 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 $X_1,...,X_N$ have an interpretation as the ``sufficient digits'', since the distribution of $R=(X_{N+1},X_{N+2},...)$ conditioned on $S=(X_1,...,X_N)$ 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 $s\ge0$. If $s=0$ then $X_{N+1},X_{N+2},...$ 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 $\max\{N,s\}+M-s$ 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 Lüroth series, $β$-expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.

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