Generalization of discrimination-rate theorems of Chernoff and Stein

We consider a simple hypothesis and alternative about an abstract random observation parametrized by A from a directed set A. The asymptotics over A is evaluated for mixed errors of the Bayes tests and second kind errors of the Neyman-Pearson tests. Similar asymptotics has been first evaluated by H. ChernofT and Ch. Stein when A is the set of naturals and the observation consists of the first X terms of a sequence of i.i.d. r.v.'s (cf. [3], [4]). Extensions to the observation consisting of segments of random sequences, processes and fields have been studied by many authors. We show that the extension essentially depends on the existence of an asymptotic Renyi distance of the hypothesis and alternative and that this distance explicitly describes the discrimination rates attained by the Bayes and Neyman-Pearson tests. We do not use the theory of large deviations — all results are deduced from several simple properties of Renyi distances established by Liese and Vajda in [20] and from elementary inequalities established for Renyi distances by Kraft and Plachky and Vajda (cf. [15], [31]).