Lower Bound for Error Probability in a Discrete Memoryless Channel with Feedback

It is shown that in the case of block transmission of length $N$ with feedback over a discrete memoryless channel, the error probability satisfies the lower spherical-packing bound at rates less than the channel capacity, while the probability of correct reception satisfies the Arimoto upper bound at rates greater than the capacity. Thus, the logarithmic asymptotic behavior (as $N\to\infty$) of the error probability and probability of correct reception is known at least for all rates greater than the critical one.