Uniform Continuity in Distribution

• Uniform continuity in distribution is a framework that examines the stability of probabilistic functionals and transformations under perturbations of input measures.
• It employs metrics like total variation and Lévy–Prokhorov distances to quantify error bounds in entropy and ensure robust transformations of densities.
• Applications include Gaussian multiplicative chaos, variational CDF path length minimization, and finite-dimensional reductions that support advanced probabilistic analysis.

Uniform continuity in distribution refers to stability properties of probabilistic functionals and transformations under perturbations of input measures or maps. It plays a central role in modern probability theory, stochastic process analysis, and the study of information-theoretic quantities such as differential entropy. Three distinct but connected paradigms dominate the current literature: uniform continuity of entropy functionals on classes of densities under total variation; geometric uniformity via CDF path length minimization; and uniform continuity in distribution of Borel transformations of random fields, analyzed via convergence in the space of probability measures metrized by Lévy–Prokhorov distance.

1. Classes of Distributions for Uniform Continuity of Differential Entropy

A rigorous framework for establishing uniform continuity of the differential entropy functional involves restricting attention to the parameterized class $(\alpha,v,m)\text{–}\mathcal{AC}^n$ (Ghourchian et al., 2017). A probability measure $P$ on $\mathbb{R}^n$ is included in this class if it is absolutely continuous with respect to Lebesgue measure, with a density $p(x)$ satisfying

• Density bound: $\mathrm{ess}\,\sup_{x\in\mathbb{R}^n} p(x) < m$,
• Moment bound: $$\int_{\mathbb{R}^n} \|x\|_\alpha^\alpha\,p(x)\,dx < v$$, where $$\|x\|_\alpha = (\sum_{i=1}^n |x_i|^\alpha)^{1/\alpha}$$ for some fixed $\alpha > 0$, $v > 0$, $m > 0$. These requirements can be verified for standard probability distributions such as Gaussian, exponential, and uniform densities by direct calculation of their moments and supremum bounds.

2. Metrics Governing Uniform Continuity

The principal metrics of interest include:

• Total variation distance: $\|p-q\|_1 := \int_{\mathbb{R}^n} |p(x)-q(x)|\,dx$, equivalently $d_{TV}(p,q) = \frac{1}{2}\|p-q\|_1$.
• Relative entropy (Kullback–Leibler divergence): $D(p\Vert q) = \int p(x)\log\frac{p(x)}{q(x)}\,dx$, relevant in separate literature strands but not used for the quantitative uniform continuity bound in (Ghourchian et al., 2017).
• Wasserstein distance: Used more generally for weak convergence, but not fundamental for the principal continuity result in the referenced work.

Uniform continuity of entropy is quantitatively controlled with respect to total variation.

3. Uniform Continuity of Differential Entropy: Theorem and Quantitative Bound

The central result is a uniform continuity theorem for the differential entropy functional $h(p) = -\int p(x)\log p(x)\,dx$ defined on $(\alpha,v,m)\text{–}\mathcal{AC}^n$ (Ghourchian et al., 2017). Specifically, for densities $p,q$ in this class, with $\delta = \|p-q\|_1 \le m$, the following holds: $$|h(p)-h(q)| \le c_1\,\delta + c_2\,\delta\,\log\left(\frac{1}{\delta}\right),$$ where $c_1$ and $c_2$ are explicit constants depending only on $(\alpha,v,m,n)$. As $\delta \to 0$, the bound vanishes, establishing uniform continuity of $h$ with respect to total variation.

The proof combines rescaling, comparison with generalized normal densities, convexity inequalities for the $x \log x$ functional, and careful finite-dimensional reduction to control the entropy difference via a signed difference density $z(x) = |p(x)-q(x)|/\delta$, itself parametrized in $(\alpha,v',m')\text{–}\mathcal{AC}$.

4. Geometric Quantification of Uniformity: CDF Path Length Minimization

An alternative (and geometrically motivated) perspective is offered via minimization of the cumulative distribution function (CDF) path length (Beyer, 2015). For a differentiable CDF $F$ on $[a,b]$, the arc length is

$$L[F] = \int_a^b \sqrt{1 + (F'(x))^2}\,dx = \int_a^b \sqrt{1 + f(x)^2}\,dx,$$

where $f(x) = F'(x)$ is the density. The uniform distribution, with $F_U(x) = \frac{x-a}{b-a}$, uniquely minimizes $L[F]$. Imposing raw-moment constraints leads to "shortest-path" distributions (SPDs) via a variational Euler–Lagrange framework: $$f(x) = \frac{\lambda_0 + \lambda_1 x + \lambda_2 x^2}{\sqrt{1 - (\lambda_0 + \lambda_1 x + \lambda_2 x^2)^2}},$$ with multipliers $\{\lambda_i\}$ determined by moment equations. Numerical and analytical studies confirm that SPDs spontaneously induce heavier tails than maximum-entropy (ME) distributions, emphasizing that path length and entropy capture distinct aspects of uncertainty. The path-length criterion thus provides a geometric measure of (non-)uniformity and by extension, uniform continuity in distribution.

5. Uniform Continuity in Distribution for Borel Transformations of Random Fields

Uniform continuity in distribution can be rigorously established for Borel transformations of random fields, with formal conditions involving compactness, full support, and Borel measurability (Bufetov, 30 Dec 2025). In this framework, the state space $(W,d)$ is typically a complete separable metric space; probability measures are drawn from compact subsets $\mathcal{A} \subset \mathcal{P}_1(W)$ and images collected in $\widetilde{\mathcal{A}} \subset \mathcal{P}_1(W)$.

The key definition: a family $\{X_t:t\in T\}$ of random elements is uniformly continuous in distribution (w.r.t.\ a metric $d_T$ on $T$) if, for every $\varepsilon>0$, there exists $\delta>0$ such that $d_T(s,t) < \delta$ implies $$d_{\rm LP}(\mathbb{P}\circ X_s^{-1}, \mathbb{P}\circ X_t^{-1}) < \varepsilon$$, where $d_{\rm LP}$ is the Lévy–Prokhorov distance.

The main theorem asserts that the map

$$g \mapsto g_*:\; B_{\widetilde{\mathcal{A}}}(\mathcal{A}\times V, W) \to B(\mathcal{A}, \widetilde{\mathcal{A}})$$

is uniformly continuous under metrics of uniform convergence in probability (input) and Tchebycheff–uniform (output). The proof strategy leverages finite-dimensional reductions, compactness, and coordinate-wise convergence, demonstrating stability of distributional outputs under perturbations of input mappings.

6. Illustrative Example and Intuitive Implications

A canonical example is Gaussian multiplicative chaos (GMC), where the exponential of an indexed random field is normalized and pushed forward as a measure (Bufetov, 30 Dec 2025). Uniform continuity follows provided moment bounds ensure compactness of input law-sets, and the input map is Borel measurable. This guarantees that approximation schemes for GMC, or for broader classes of random fields, inherit distributional stability—a key requirement for rigorous probabilistic analysis and simulation algorithms.

The synthesis is: tightness (compactness) prevents mass escape, finite-dimensional truncations enable error control, and convergence metrics (Lévy–Prokhorov, total variation) enforce uniformity of transformation. These principles underlie the stability of distributional laws in complex stochastic systems.

7. Comparison, Contrasts, and Further Research Directions

Uniform continuity in distribution intersects with maximum-entropy methods, geometric criteria (path length minimization), and entropy stability. The empirical fact that shortest-path distributions can exhibit heavier tails than maximum-entropy analogues (Beyer, 2015) highlights divergent risk profiles and distributional behaviors under the two paradigms.

A plausible implication is that uniform continuity in distribution can be quantified both analytically (via entropy bounds with respect to total variation) and geometrically (via CDF arc length minimization), and that the choice of metric and constraint set critically affects both explicit continuity bounds and qualitative properties of the resulting distributions.

Future work may address extension to non-Euclidean state spaces, multi-moment constraint schemes, and granular analysis of distributional stability for non-linear stochastic transformations. Ongoing research continues to elaborate tightness conditions, convergence metrics, and uniform continuity results for broader classes of functionals and maps in probability theory and statistical mechanics.