A homogenization principle for total variation

Abstract: A homogenization principle for total variation We prove an inequality comparing the variational distance between pairs of product probability measures to its homogenized counterpart. If $P_1,\ldots,P_n,Q_1,\ldots,Q_n$ are arbitrary probability measures on a measurable space and $\bar P:=\frac1n\sum_{i=1}^n P_i, \bar Q:=\frac1n\sum_{i=1}^n Q_i $, we show that $$TV!\left(\bigotimes_{i=1}^n P_i, \bigotimes_{i=1}^n Q_i\right) \;\ge\; c\,TV(\bar P^{\otimes n},\bar Q^{\otimes n}),$$ where $c>0$ is a universal constant. The proof is based on a one-dimensional representation of total variation between products. We embed pairs of probability distributions $P_i,Q_i$ into positive measures $ηi$ on $\mathbb{R}$. We then define a functional $T$ over measures on $\mathbb{R}$ that realizes TV over products via convolution: $TV!\left(\bigotimes{i=1}^n P_i, \bigotimes_{i=1}^n Q_i\right)=T(η1*\cdots η_n)$. Our main analytic discovery is that for the relevant class of positive measures $η_i$, the convolution inequality $T(η_1\cdots*η_n) \ge c\,T!\left(\barη^{*n}\right)$ holds, where $\barη=\frac1n\sum{i=1}^n η_i$. Finally, a higher-dimensional lifting argument shows that $T!\left(\barη^{*n}\right)\ge TV(\bar P^{\otimes n},\bar Q^{\otimes n})$. To our knowledge, both the exact representation and the convolution inequality are new.

• The paper introduces a homogenization principle that bounds the total variation distance between product measures using a universal constant.
• It employs a novel convolutional encoding of probability measures to represent TV distances as convolution functionals.
• The work provides theoretical insights with practical implications for statistical testing and high-dimensional probability analysis.

A Homogenization Principle for Total Variation: Summary and Implications

Overview

This paper establishes a novel and general lower bound for the total variation (TV) distance between product measures. Specifically, if $P_1, \ldots, P_n$ and $Q_1, \ldots, Q_n$ are arbitrary probabilities on a measurable space, it is shown that the TV distance between their product measures dominates, up to an explicit universal constant, the TV distance between the product measures constructed from the mean distributions: $\TV \left( \bigotimes_{i=1}^n P_i,\, \bigotimes_{i=1}^n Q_i \right) \geq c\, \TV \left( \bar{P}^{\otimes n},\, \bar{Q}^{\otimes n} \right)$ with $c > 0.1489$. Here, $\bar{P} = \frac{1}{n} \sum_{i=1}^n P_i$, $\bar{Q} = \frac{1}{n} \sum_{i=1}^n Q_i$. The result operates over all measurable spaces, with the constant $c$ being independent of the specifics of the distributions or the underlying space.

Analytic Framework and Technical Contribution

The central theoretical contribution is an exact one-dimensional representation of TV distance between product measures in terms of a convolution functional over positive measures. For probability mass functions $P_i, Q_i$ on a finite set $\Omega$, an encoding is introduced: $$\eta_i := \sum_{\omega \in \Omega} \sqrt{P_i(\omega) Q_i(\omega)}\, \delta_{\frac{1}{2} \log \frac{P_i(\omega)}{Q_i(\omega)}}$$ with a functional $Q_1, \ldots, Q_n$0 defined via

$Q_1, \ldots, Q_n$1

It is established that the TV between product measures linearizes: $Q_1, \ldots, Q_n$2 The main analytic innovation is a comparison ("homogenization principle") showing that: $Q_1, \ldots, Q_n$3 where $Q_1, \ldots, Q_n$4 and $Q_1, \ldots, Q_n$5 is the (convolution) operation. Lifting arguments then translate back from the representation space to the original measure-theoretic setting, yielding the lower bound of interest for the original TV quantities.

Both the exact convolutional representation and the specific convolution inequality are new. The proof employs a multilinear analysis of functionals over random variables constructed from the admissible measures, with detailed analysis partitioned into small and large mass defect regimes.

Noteworthy Results and Optimality

• Explicit Constant: The universal constant is quantified as $Q_1, \ldots, Q_n$6 (or equivalently $Q_1, \ldots, Q_n$7). For Bernoulli products, previous work showed $Q_1, \ldots, Q_n$8 and conjectured this upper bound to be tight ("TV homogenization inequalities," (Kontorovich, 7 Jan 2026)). Precise optimal values and maximizing measures remain open.
• Special Case: For finite $Q_1, \ldots, Q_n$9, the homogenized TV distance reduces to difference between corresponding multinomial distributions:

$\TV \left( \bigotimes_{i=1}^n P_i,\, \bigotimes_{i=1}^n Q_i \right) \geq c\, \TV \left( \bar{P}^{\otimes n},\, \bar{Q}^{\otimes n} \right)$0

• No Reverse Inequality: The lower bound is not reversible; counterexamples demonstrate that the heterogeneous TV can be much larger.

Contextualization and Prior Work

The question of comparing the heterogeneous product TV to its homogenized version is natural due to the analytic tractability of the homogeneous case (governed by the Chernoff information asymptotics). In contrast, the inhomogeneous case is much less tractable. Results from Kakutani, Hellinger, and Bhattacharyya inform the representation of product measures, but the approach in this work is distinguished by its precise nonlinear encoding and convolution-based analysis.

Previous TV lower bounds based on $\TV \left( \bigotimes_{i=1}^n P_i,\, \bigotimes_{i=1}^n Q_i \right) \geq c\, \TV \left( \bar{P}^{\otimes n},\, \bar{Q}^{\otimes n} \right)$1 structure (e.g., [kon25tens]) are generally incomparable and do not yield a homogenization principle. Approximations and bounds for convolutions of probability measures, as explored in [Roos2010ClosenessConvolutions, Roos2017RefinedTV], address different analytic regimes and do not recover the present result.

Implications and Future Directions

Theoretical Implications

This homogenization bound provides a direct quantitative link between arbitrary products and their mean-product analogs in TV distance. Such bounds are fundamental to high-dimensional probability theory, information theory, and the theory of statistical experiments, particularly in the analysis of product measures arising in composite hypothesis testing or mutual information estimation for mixtures.

The convolutional encoding and the emergence of TV as a functional thereof highlight new structural aspects of divergence measures, revealing connections with Hellinger geometry and bounded transforms used in robust statistics.

Practical Considerations

On the algorithmic and statistical side, the result enables practitioners to bound the TV distance of a product of arbitrary distributions via the more tractable case of product means, which can be significant, for example, in distribution learning or testing with heterogeneous data sources. This result also ensures a nontrivial lower bound for the TV contrast between composite models.

Prospects for Future Research

• Optimal Constants: Determining the exact universal constant $\TV \left( \bigotimes_{i=1}^n P_i,\, \bigotimes_{i=1}^n Q_i \right) \geq c\, \TV \left( \bar{P}^{\otimes n},\, \bar{Q}^{\otimes n} \right)$2, as well as identifying extremizers (measures achieving the bound), remains open.
• Extensions to Other Divergences: The approach suggests analogs for other $\TV \left( \bigotimes_{i=1}^n P_i,\, \bigotimes_{i=1}^n Q_i \right) \geq c\, \TV \left( \bar{P}^{\otimes n},\, \bar{Q}^{\otimes n} \right)$3-divergences, particularly those with convolution-friendly representations.
• Applications to High-dimensional Asymptotics: Leveraging this homogenization principle in asymptotic regime analysis for high-dimensional composite testing and learning theory problems is a natural direction.
• Algorithmic Ramifications: The structural representations may spur new algorithmic methods for deterministic or approximate calculation of TV distances between product measures, see also [Feng24Deterministically].

Conclusion

This work introduces and proves a homogenization principle for total variation, relating the TV between general product measures to their mean-product forms via convolutional analytic techniques. The result holds with an explicit universal constant and applies in complete generality. The representation and analytic framework open new directions for the study of product measures, divergence bounds, and their applications across probability, information theory, and statistics.