Correction to "The Generalized Stochastic Likelihood Decoder: Random Coding and Expurgated Bounds"

In a recent article [1], random coding error exponents and expurgated exponents were analyzed for the generalized likelihood decoder (GLD), where the decoded message is randomly selected under a probability distribution that is proportional to a general exponential function of the empirical joint distribution of the codeword and the channel output vectors. In Section V of [1], Theorem 2 provides an expurgated exponent which is applicable to this decoder (and hence also to the optimal maximum likelihood decoder). The proof of that theorem is based on two steps of a certain expurgation procedure. Nir Weinberger has brought to my attention that there is a certain gap in that proof, as the second expurgation step might interfere with the first step (more details will follow in Section 2 of this letter). The purpose of this letter is to provide an alternative proof to the above mentioned theorem.