Tightness of the Asymptotic Generalized Poor-Verdú Error Bound for the Memoryless Symmetric Channel

The generalized Poor-Verdu error lower bound for multihypothesis testing is revisited. Its asymptotic expression is established in closed-form as its tilting parameter grows to infinity. The asymptotic generalized lower bound is then studied in the classical channel coding context where it is proved that for any sequence of block codes sent over the memoryless binary symmetric channel (BSC), the minimum probability of decoding error has a relative deviation from the generalized bound that grows at most linearly in blocklength. A direct consequence of this result is that the asymptotic generalized bound achieves the error exponent (or reliability function) of the BSC at arbitrary coding rates. Finally, these tightness results are extended for the class of memoryless non-binary symmetric channels.