On the asymptotics of M-hypothesis Bayesian detection

In two-hypothesis detection problems with i.i.d. observations, the minimum error probability decays exponentially with the amount of data, with the constant in the exponent equal to the Chernoff distance between the probability distributions characterizing the hypotheses. We extend this result to the general M-hypothesis Bayesian detection problem where zero cost is assigned to correct decisions, and find that the Bayesian cost function´s exponential decay constant equals the minimum Chernoff distance among all distinct pairs of hypothesized probability distributions