Multivariate majorization of continuous statistical experiments

Abstract: We derive sufficient and almost necessary conditions for large sample and catalytic majorization between finite statistical experiments over standard Borel sample spaces. This work generalizes previous results, on one hand, in the bivariate case and, on the other hand, in the multivariate discrete (or, rather, finite) case, i.e., matrix majorization. We derive multivariate generalizations of the bivariate Renyi relative entropies and show that inequalities involving these multivariate Renyi divergences characterize large-sample and catalytic majorization of finite statistical experiments. As our methods are real-algebraic in nature, this work demonstrates that large deviation techniques are not the only option available to derive conditions for large sample majorization even in the case of more general sample spaces of the experiments. We also show that all general multivariate divergences, i.e., multivariate extensive and monotone maps of finite statistical experiments, can be expressed through barycentres over the set of multivariate Renyi divergences. We also show that we may characterize the optimal conversion rate of a statistical experiment into another using the multivariate Renyi divergences.

• The paper introduces multivariate Rényi divergences as key invariants for information convertibility between statistical experiments.
• It employs a real‐algebraic approach to derive almost necessary conditions for large-sample and catalytic majorization.
• It provides explicit operational rates for transforming experiments and outlines implications for quantum statistics and resource theories.

Multivariate Majorization of Continuous Statistical Experiments: A Technical Analysis

Introduction

This paper develops a real-algebraic framework for characterizing informational convertibility between finite statistical experiments over general (standard Borel) spaces, focusing on multivariate majorization in both large-sample and catalytic regimes. By systematically generalizing results previously established for bivariate and finite-matrix settings, the work introduces multivariate Rényi divergences as the central quantities determining majorization relations. The algebraic, as opposed to large-deviation, approach presented here yields both sufficient and "almost necessary" conditions, advances the structure theory of divergences in statistical experiments, and offers explicit formulas for optimal conversion rates between experiments in terms of these divergences.

Majorization of Statistical Experiments: Definitions and Notational Regime

A statistical experiment is formalized as a tuple $P = \{p^{(k)}\}_{k=1}^d$ of probability measures indexed by a finite parameter set $\Theta$, all dominated by a common measure class with uniformly bounded Radon-Nikodym derivatives. Majorization $P \succeq Q$ between experiments is defined via the existence of a Markov kernel transforming the tuple of $P$'s distributions into those of $Q$. Large-sample majorization refers to majorization of $n$-fold products for sufficiently large $n$, while catalytic majorization admits the use of an ancillary statistical experiment ("catalyst") such that a joint product can be majorized.

Crucially, the algebra centers on a preordered semiring $S^d$ constructed from equivalence classes of regular $d$-tuples, equipped with operations mimicking disjoint union ($+$) and tensor products ($\Theta$0), and a preorder encoding majorization. This abstract algebraic structure enables direct application of asymptotic spectra techniques.

Real-Algebraic Methods and the Vergleichsstellensatz

The paper exploits foundational results in the theory of preordered semirings, particularly a Vergleichsstellensatz (comparison theorem), which characterizes ordering in terms of inequalities over a well-defined test spectrum of monotone semiring homomorphisms and derivations. The test spectrum encodes, in analytic form, the operational monotonic quantities (divergences) relevant to stochastic transformations between statistical experiments. This allows for algebraic (as opposed to more conventional large deviations) derivation of informational conversion criteria.

Multivariate Rényi Divergences

A central contribution is the rigorous construction and deployment of multivariate Rényi divergences $\Theta$1, parametrized by vectors $\Theta$2 (satisfying $\Theta$3 and additional constraints). For a statistical experiment $\Theta$4, these divergences are defined as

$\Theta$5

for appropriate $\Theta$6. These divergences subsume both classical Rényi and Kullback-Leibler (KL) divergences as specific cases and their pointwise limits recover so-called tropical divergences and derivations (linear combinations of pairwise KL divergences).

The paper demonstrates that temperate and tropical multivariate Rényi divergences are exhaustive for characterizing the asymptotic convertibility of statistical experiments, both as order parameters for large-sample/catalytic majorization and as the building blocks of all tensor-additive, monotone divergences.

Main Results

Large-Sample and Catalytic Majorization Criteria

The principal theorem gives sufficient and nearly necessary conditions for $\Theta$7 to majorize $\Theta$8 in the large-sample (and thus also the catalytic) regime: if and only if

• $\Theta$9 for all admissible $P \succeq Q$0 (temperate divergences),
• $P \succeq Q$1 for all $P \succeq Q$2 (tropical divergences),
• $P \succeq Q$3 for all $P \succeq Q$4 (KL derivations),

then $P \succeq Q$5 for $P \succeq Q$6 sufficiently large, and a catalyst $P \succeq Q$7 exists such that $P \succeq Q$8. The conditions are also shown to be necessary in a relaxed (non-strict) form.

Algebraic Structure and Uniqueness of Divergences

A comprehensive result on the structure of divergences is established: Every additive, monotone map (divergence) on the space of statistical experiments can be written as a barycentric combination (integral) over the multivariate Rényi divergences (including certain limits):

$P \succeq Q$9

for some positive measures $P$0, $P$1, and coefficients $P$2. This characterization is achieved using functional analysis (Kantorovich and Riesz-Markov theorems) applied within the asymptotic spectra formalism. The result is equivalent to closely related findings using large deviation and measure-theoretic tools [see e.g., Theorem 5.1 in "balsubramani2026"].

Optimal Rates of Statistical Experiment Conversion

For two experiments $P$3 and $P$4 with pairwise distinct measures, the optimal rate $P$5 at which $P$6 can be transformed into $P$7 is characterized by

$P$8

where the infimum is taken over all admissible $P$9. This generalizes and strengthens prior results on bivariate and finite-outcome cases, providing an explicit operational formula grounded in the Rényi divergence landscape.

Numerical and Structural Highlights

• The inequalities formulating large-sample and catalytic majorization are sharp: strict inequalities yield sufficiency; necessity holds for equality.
• The representation theorem for divergences is exhaustive: no further monotone, tensor-additive functions exist beyond barycenters of multivariate Rényi divergences and their limits.
• The algebraic real-analysis framework allows one to forgo large deviation techniques, leveraging only measure-theoretic and semiring properties.

Implications and Future Directions

This work systematically resolves the structure of majorization and divergence for finite statistical experiments over general sample spaces, establishing a clear bridge between algebraic spectra, operational convertibility, and information measures. The results are expected to influence:

• Quantum statistics (where experiments correspond to POVM tuples over standard Borel spaces)
• Asymptotic resource theories in information theory
• Functional analysis of statistical morphisms
• The design of optimal information-preserving transformations between statistical models and tests

Potential extensions include removing or relaxing the boundedness condition (mutual absolute continuity and essential boundedness of Radon-Nikodym derivatives) required for the main theorems, as well as developing parallel results for broader support conditions or infinite parameter sets.

Conclusion

Through a unified real-algebraic framework, the paper provides a rigorous, explicit, and general blueprint for understanding statistical majorization in the multivariate, continuous setting. Multivariate Rényi divergences are established as fundamental invariants governing the convertibility of information in statistical experiments, with all divergences and rates deriving from their structure. This extends and synthesizes lines of inquiry previously pursued via disparate large deviation and matrix-theoretic techniques. The algebraic approach outlined here is likely to influence future technical work on resource interconversion, operational information theory, and quantum statistical decision theory.