  On a compact group the Haar probability measure plays the role of uniform
distribution. The entropy and rate distortion theory for this uniform
distribution is studied. New results and simplified proofs on convergence of
convolutions on compact groups are presented and they can be formulated as
entropy increases to its maximum. Information theoretic techniques and Markov
chains play a crucial role. The convergence results are also formulated via
rate distortion functions. The rate of convergence is shown to be exponential.
