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

Let {Yn} be a stationary random process where each Y value takes values in a finite set B. Then define Hn and en so that H is the entropy of {Yn} 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 {Hn - H} and {en - e} are not, in general, asymptotically identical for a stationary process {Yn}.