Extended Continuous Mapping Theorem

• The extended continuous mapping theorem is a framework that generalizes classical convergence principles to encompass piecewise continuous functions and random processes with discontinuities.
• It introduces convergence with restraint, ensuring robust control of suprema and accurate asymptotic bounds even when uniform convergence fails.
• The theorem underpins advanced analyses in spatial statistics and topological data analysis, enabling valid confidence regions for complex set functionals.

The extended continuous mapping theorem (CMT) generalizes the classical functional convergence principles to encompass sequences of piecewise continuous functions and random processes whose limit objects may exhibit discontinuities or partition-dependent jump behavior. The concept, as developed in (Maullin-Sapey et al., 11 Nov 2025), is characterized by the introduction of convergence "with restraint," which ensures robust convergence of suprema and maxima functionals even in absence of uniform convergence—a property crucial for contemporary statistical applications in spatial statistics, topological data analysis, and the study of excursion set uncertainty. The theorem provides two-sided probabilistic bounds for functionals of converging sequences and prescribes sufficient conditions for transformation maps (including maxima, unions, and more complex set functionals) to preserve asymptotic behaviors.

1. Motivation and Historical Limitations

Classical limit theorems, including the original CMT, depend heavily on uniform convergence or pointwise convergence of continuous functions. Such frameworks break down when analyzing spatial processes, imaging data, or random fields with piecewise continuity—common in neuroimaging, climatology, and cosmology—where the limit exhibits jump discontinuities along a fixed partition of the domain. Traditional topologies (e.g., Skorokhod J1/M1, uniform topology on function spaces) cannot capture correct convergence behavior for suprema or level-set statistics when discontinuities are correlated with geometric structures. Pathological behaviors such as "overshoots" near jumps or non-convergent suprema motivate the need for the restrained-convergence formalism.

2. Definitions: Piecewise Continuity and Convergence with Restraint

Let $S$ be a compact metric space decomposed into a locally finite partition $\{V_i\}_{i\in I}$, and denote by $P(\{V_i\})$ the space of functions $f:S\to\mathbb{R}$ admitting continuous extensions $f_i$ to the closure $\overline V_i$ of each partition cell. A sequence $\{f_n\}$ is said to converge to $f$ with upper restraint if:

• $f_n(s)\to f(s)$ pointwise for all $s\in S$;
• For each $i$ and $s\in V_i$,

$$\lim_{\delta\downarrow 0} \limsup_{n\to\infty} \sup_{s'\in B(s,\delta)\cap V_i} f_n(s') \leq \max\{f_i(s),f_i(s)\}$$

where $f_i(s)$ refers to the one-sided extension at a boundary point. If the corresponding condition holds for $-f_n$, we have lower restraint; if both, we write $f_n\to_r f$ (Editor's term: restrained convergence). Uniform convergence implies restrained convergence, but not conversely.

3. Supremum Bounds and Pointwise-Restrained Convergence

A fundamental result establishes that under pointwise convergence, liminf of suprema over a sequence of sets $A_n$ is lower bounded by $\sup_{s\in A_\infty} f(s)$, with $A_\infty=\liminf_{n\to\infty}A_n$. Control of the limsup requires the restrained-bound property. Specifically, for functions $f_n\to_r f$ subordinate to partition $\{V_i\}$ and any sequence of sets $A_n\subset S$,

$$\limsup_{n\to\infty} \sup_{s\in A_n} f_n(s) \leq \max_{i\in I} \sup_{s\in A_\infty^+} f(s)$$

where $$A_\infty^+=\bigcap_{k=1}^\infty\,\overline{\cup_{n\geq k}A_n\cap V_i}$$ for each $i$.

This integrative control is critical for the analysis of maxima, excursion sets, and union/intersection statistics of spatial processes when uniform convergence cannot hold.

4. Suprema-Preserving Maps and Extended CMT

A map $H: F\to P(\{V_i\})$ is suprema-preserving if, given any uniformly convergent $f_n\to f$ in $F$, $H(f)$ is an upper restrained bound for $\{H(f_n)\}$. Sufficient conditions include:

• There exists $\alpha>0$ such that $$\|H(f_n)-H(f)\|_\infty = O(\|f_n-f\|_\infty^\alpha)$$;
• For each $i, s\in V_i$, $$\lim_{\delta\downarrow 0}\limsup_{n\to\infty} \sup_{s'\in B(s,\delta)\cap V_i} [H(f_n)(s')-H(f)(s')] \leq \max\{0, H(f_i)(s)-H(f_i)(s)\}$$.

These criteria enable deterministic and probabilistic functionals such as maxima, unions, and intersections to inherit convergence properties via transformation, thus extending the continuous mapping theorem to restrained limits.

5. Extension to Random Processes and Probabilistic Suprema Bounds

Given random processes $G_n: \Omega\to \ell^\infty(S)$ and $G=\lim G_n$, assume $G_n=H\circ X_n$ for some suprema-preserving $H$ and uniformly convergent $X_n$. Then for deterministic sets $A_n, B_n\subset S$,

$$\limsup_{n\to\infty} \mathbb{P}^*\left(\max\left\{\sup_{s\in A_n} G_n(s), \sup_{s\in B_n} G_n(s)\right\} \leq q\right) \leq \mathbb{P}\left( \max\left\{ \sup_{s\in A_\infty} G(s), \sup_{s\in B_\infty^+} G(s) \right\}\leq q \right)$$

and

$$\liminf_{n\to\infty} \mathbb{P}_*\left(\max\left\{\sup_{s\in A_n} G_n(s), \sup_{s\in B_n} G_n(s)\right\}<q\right) \geq \mathbb{P} \left( \max\left\{ \sup_{s\in A_\infty^+} G(s), \sup_{s\in B_\infty} G(s)\right\}<q \right)$$

for outer/inner probabilities $\mathbb{P}^*,\mathbb{P}_*$ respectively. The two-sided bounds are tight when the restrained-bound conditions are uniformly satisfied.

6. Illustrative Examples and Limitations

Canonical examples demonstrate that pointwise convergence alone is insufficient for suprema convergence at partition boundaries and that artificially constructed "bumps" can violate restraint requirements. Restrained convergence precisely excludes these pathologies, while uniform convergence remains a sufficient but not necessary condition. Notably, for functionals not representable as piecewise continuous maps (e.g., symmetric differences with singular boundaries), the theory requires confinement pairs (brackets) to recover interval bounds rather than exact convergence.

7. Implications, Generalizations, and Applications

The extended CMT and restrained convergence formalism are specifically tailored to modern challenges in spatial statistics (as in (Maullin-Sapey et al., 11 Nov 2025)), uncertainty quantification, and topological data analysis of random fields with partition-dependent structure. The theory enables construction of valid confidence regions and asymptotic results for general maxima, unions, intersections, and more complex set functionals of imaging data, even when sample paths admit non-differentiable behavior. Hybridization with existing CLT results and mapping theorems provides a flexible, partition-aware foundation for the rigorous study of excursion set uncertainty, bridging functional analysis, probability theory, and applied statistical methodology.

In summary, the extended continuous mapping theorem governed by convergence with restraint forms a robust and comprehensive framework for the asymptotic analysis of piecewise continuous stochastic and deterministic systems, resolving the limitations of both classical uniform-convergence approaches and function-space topologies in high-dimensional or geometrically-structured domains (Maullin-Sapey et al., 11 Nov 2025).