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