A doubly composite Chernoff-Stein lemma and its applications
Ludovico Lami
TL;DR
The paper addresses the problem of hypothesis testing when both null and alternative are composite and genuinely correlated, introducing a generalised doubly composite Chernoff–Stein lemma. It develops a symbol-by-symbol blurring technique and a meta-lemma to bound type-II error exponents via regularised relative entropies, yielding a formula $\mathrm{Stein}(\mathcal{R}\|\mathcal{S}) = \inf_{P\in \mathcal{R}_1} D^{\infty}(P\|\mathrm{conv}(\mathcal{S}))$, with single-letter simplifications in cases where the alternative is composite i.i.d. or arbitrarily varying. The contribution also includes a weakened set of axioms that avoid full permutational symmetry, and a constrained de Finetti reduction framework connecting symmetric tests to convex combinations of i.i.d. distributions. Overall, the results unify and strengthen many classical and quantum-inspired Stein and Sanov-type theorems, extend applicability to memoryful sources, and provide practical single-letter expressions in key settings, with broad implications for information theory and statistical inference. The methods combine refined combinatorial bounds, per-symbol noise addition, and entropy-based arguments to achieve sharp asymptotics in high-dimensional hypothesis testing problems. This work paves the way for tighter classical and quantum hypothesis-testing results under general convex constraints and correlation structures.
Abstract
Given a sequence of random variables $X^n=X_1,\ldots, X_n$, discriminating between two hypotheses on the underlying probability distribution is a key task in statistics and information theory. Of interest here is the Stein exponent, i.e. the largest rate of decay (in $n$) of the type II error probability for a vanishingly small type I error probability. When the hypotheses are simple and i.i.d., the Chernoff-Stein lemma states that this is given by the relative entropy between the single-copy probability distributions. Generalisations of this result exist in the case of composite hypotheses, but mostly to settings where the probability distribution of $X^n$ is not genuinely correlated, but rather, e.g., a convex combination of product distributions with components taken from a base set. Here, we establish a general Chernoff-Stein lemma that applies to the setting where both hypotheses are composite and genuinely correlated, satisfying only generic assumptions such as convexity (on both hypotheses) and some weak form of permutational symmetry (on either hypothesis). Our result, which strictly subsumes most prior work, is proved using a refinement of the blurring technique developed in the context of the generalised quantum Stein's lemma [Lami, IEEE Trans. Inf. Theory 2025]. In this refined form, blurring is applied symbol by symbol, which makes it both stronger and applicable also in the absence of permutational symmetry. The second part of the work is devoted to applications: we provide a single-letter formula for the Stein exponent characterising the discrimination of broad families of null hypotheses vs a composite i.i.d. or an arbitrarily varying alternative hypothesis, and establish a 'constrained de Finetti reduction' statement that covers a wide family of convex constraints. Applications to quantum hypothesis testing are explored in a related paper [Lami, arXiv:today].