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= (X₍+₁, X₍+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 s0. If s=0 then X₍+₁, X₍+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 \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 L\"uroth series, -expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.