On the asymptotic behavior of the approximations for the probability of error and the entropy of a stationary random process

Let {Y_n} be a stationary random process where each Y value takes values in a finite set B. Then define H_n and e_n so that H is the entropy of {Y_n} and e is the probability of error in predicting Y₀ from the infinite past. The purpose of this correspondence is to give an example which shows that {H_n - H} and {e_n - e} are not, in general, asymptotically identical for a stationary process {Y_n}.