Convergence in Distribution of the Error Exponent of Random Codes at Zero Rate

We study the convergence in distribution of the error exponent of random codes, defined as the negative normalized logarithm of the probability of error, of both i.i.d. and constant-composition ensembles over discrete memoryless channels. For a constant number of messages, the distribution of the error exponent converges to that of the minimum of a set of independent normal random variables. For an increasing sub-exponential number of messages, the error exponent converges to a normal distribution, independent of the number of messages. As a byproduct, we provide a new method to prove the convergence to a normal distribution of an infinite number of random variables based on a modification of the Wasserstein metric.