Combinatorial Characterization of Sequences with Bounded Variance

Abstract

In many contexts, a necessary condition for an asymptotic limit theorem is the variability condition, that is the variance has to be unbounded. We give a combinatorial characterization of transducers whose output sum does not satisfy this variability condition: The output of each cycle of the transducer has to be proportional to its length. ??? This characterization allows to prove a central limit theorem for a sequence for which there exists a transducer that it is too large or not explicitly given. For example, we can prove that the variability condition is satisfied for the optimal Hamming weight of many $\tau$-adic digit expansions for an algebraic number $\tau$.