  We study the joint distribution of the input sum and the output sum of a
deterministic transducer. Here, the input of this finite-state machine is a
uniformly distributed random sequence.
  We give a simple combinatorial characterization of transducers for which the
output sum has bounded variance, and we also provide algebraic and
combinatorial characterizations of transducers for which the covariance of
input and output sum is bounded, so that the two are asymptotically
independent.
  Our results are illustrated by several examples, such as transducers that
count specific blocks in the binary expansion, the transducer that computes the
Gray code, or the transducer that computes the Hamming weight of the width-$w$
non-adjacent form digit expansion. The latter two turn out to be examples of
asymptotic independence.
