On Stein’s lemma in hypotheses testing in general non-asymptotic case

The problem of testing two simple hypotheses in a general probability space is considered. For fixed type-I error probability, the best exponential decay rate type-II investigated. In regular asymptotic cases (i.e., when length observation interval grows without limit) given by Stein’s exponent. paper, for space, some non-asymptotic lower and upper bounds are derived. These represent pure analytic relations any limiting operations. natural cases, these also give convergence Some illustrating examples provided.