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

## TL;DR

This paper investigates the optimal exponential decay rate of type-II error in hypothesis testing within general probability spaces, providing non-asymptotic bounds and insights into Stein's exponent without relying on asymptotic assumptions.

## Contribution

It derives non-asymptotic bounds for the decay rate of type-II error in hypothesis testing, extending Stein's lemma to general probability spaces without asymptotic limits.

## Key findings

- Non-asymptotic bounds for error decay rates
- Convergence rate analysis for Stein's exponent
- Illustrative examples demonstrating bounds

## Abstract

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.