# Glivenko–Cantelli theorem

The **Glivenko–Cantelli theorem** states that for independent, identically distributed real-valued random variables, the [empirical distribution function](https://www.edgepedia.org/empirical-distribution-function) converges uniformly to the true distribution function almost surely: the largest gap between the two, taken over all points at once, shrinks to zero as the sample size grows. It was proved in 1933 by [Valery Glivenko](https://www.edgepedia.org/valery-glivenko) for continuous distributions and by Francesco Cantelli for general distributions.<sup>[1](https://home.uchicago.edu/~amshaikh/webfiles/glivenko-cantelli_topics.pdf)</sup><sup> • </sup><sup>[2](https://sites.stat.washington.edu/jaw/JAW-papers/NR/jaw-gaenssler.ess.83.pdf)</sup>

| Key fact | Detail |
|---|---|
| Statement | For i.i.d. samples with distribution function F, \( \sup_{x \in \mathbb{R}} \lvert \hat{F}_n(x) - F(x) \rvert \to 0 \) almost surely<sup>[1](https://home.uchicago.edu/~amshaikh/webfiles/glivenko-cantelli_topics.pdf)</sup> |
| History | Glivenko proved it for continuous F, Cantelli for general F, both in 1933; Loeve called it the "central statistical theorem" and Renyi the "fundamental statistical theorem"<sup>[2](https://sites.stat.washington.edu/jaw/JAW-papers/NR/jaw-gaenssler.ess.83.pdf)</sup> |
| Finite-sample bound | DKW–Massart: \( P(\sup_x \lvert \hat{F}_n(x) - F(x) \rvert > \varepsilon) \le 2 e^{-2 n \varepsilon^2} \) for all \( \varepsilon > 0 \)<sup>[3](https://arxiv.org/html/2607.04387)</sup> |
| Confidence band | DKW gives a 1−δ band \( \hat{F}(t) \pm \sqrt{\log(2/\delta)/(2n)} \) valid at all points simultaneously<sup>[4](https://stephenbates19.github.io/teaching/stat_theory_2024/notes/lecture10.pdf)</sup> |
| Rate | \( \lVert F_n - F \rVert = O(n^{-1/2} (\log \log n)^{1/2}) \) with probability 1 by the law of the iterated logarithm<sup>[2](https://sites.stat.washington.edu/jaw/JAW-papers/NR/jaw-gaenssler.ess.83.pdf)</sup> |
| Class version | A class F is Glivenko–Cantelli for P if \( \sup_{f \in F} \lvert P_n f - P f \rvert \to 0 \); finite VC dimension implies it, infinite VC dimension can fail<sup>[5](https://statistique.cuso.ch/fileadmin/statistique/user_upload/3gcd.pdf)</sup><sup> • </sup><sup>[6](https://pillowmath.github.io/Expository%20Notes/VC-Dimension-and-Glivenko-Cantelli-Notes.pdf)</sup> |
| Learning theory | Uniform convergence over a hypothesis class is equivalent, for 0-1 loss, to PAC learnability and finite VC dimension<sup>[7](https://weiningchen.github.io/notes/VC_dim.pdf)</sup> |

## Statement of the theorem

Let \( X_1, \dots, X_n \) be i.i.d. random variables with distribution function F, and let the empirical distribution function be \( \hat{F}_n(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1}\{X_i \le x\} \). For each fixed x, the strong law of large numbers applied to the indicators \( \mathbf{1}\{X_i \le x\} \) gives \( \hat{F}_n(x) \to F(x) \) almost surely. The Glivenko–Cantelli theorem upgrades this pointwise statement to a uniform one: \( \sup_{x \in \mathbb{R}} \lvert \hat{F}_n(x) - F(x) \rvert \to 0 \) almost surely.<sup>[1](https://home.uchicago.edu/~amshaikh/webfiles/glivenko-cantelli_topics.pdf)</sup>

The upgrade is not automatic. A sequence of functions can converge at every fixed point while the supremum of the gap stays large: lecture notes give a spike example where \( g_n(x) \to g(x) \) for every fixed x yet \( \sup_x \lvert g_n(x) - g(x) \rvert = 1 \) for all n.<sup>[8](https://www.stat.cmu.edu/~larry/=stat705/Lecture7.pdf)</sup> [Uniform convergence](https://www.edgepedia.org/uniform-convergence) rules out such spikes, and it is what makes the empirical CDF usable as an estimator across all x at once. A notable feature is that the CDF can be estimated with no assumptions on the distribution, in contrast to density estimation, which requires strong smoothness assumptions.<sup>[8](https://www.stat.cmu.edu/~larry/=stat705/Lecture7.pdf)</sup>

## Historical origins

Both halves of the theorem appeared in 1933: Glivenko's proof covered continuous F and Cantelli's covered general F.<sup>[2](https://sites.stat.washington.edu/jaw/JAW-papers/NR/jaw-gaenssler.ess.83.pdf)</sup> Kolmogorov's 1933 work gave the limiting distribution of \( \sqrt{n} \sup_x \lvert F(x) - \bar{F}_n(x) \rvert \) as \( \sup_x \lvert B(F(x)) \rvert \) for a Brownian bridge B, and Smirnov produced a numerical table for it in 1948.<sup>[9](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/06/GlivenkoCantelli-summary.pdf)</sup> The functional central-limit extension took about twenty more years: Donsker resolved it in 1952, though not quite correctly, and Skorokhod and Kolmogorov fully corrected it in 1956; this is the Donsker theorem.<sup>[10](https://sites.stat.washington.edu/people/fanghan/teaching/STAT583/lec1.pdf)</sup>

## How the proof works

The classical proof combines the pointwise strong law with the monotonicity and ordering of the real line: a finite grid of points controls the whole function because F and \( \hat{F}_n \) are both nondecreasing. This argument relies heavily on the linear ordering of \( \mathbb{R} \), so it is hard to generalize to \( \mathbb{R}^n \), and it gives no information about the rate of convergence; [VC dimension](https://www.edgepedia.org/vc-dimension) fixes both issues.<sup>[6](https://pillowmath.github.io/Expository%20Notes/VC-Dimension-and-Glivenko-Cantelli-Notes.pdf)</sup> Symmetrization is the technique that converts finite-sample uniform deviation bounds into almost-sure convergence for finite VC classes.<sup>[6](https://pillowmath.github.io/Expository%20Notes/VC-Dimension-and-Glivenko-Cantelli-Notes.pdf)</sup>

**Measurability subtleties.** The supremum of an empirical process need not be measurable, and \( \ell_\infty[0,1] \) under the uniform metric is not even separable, so the standard theory of weak convergence does not apply. The extended theory of weak convergence developed by Dudley and Hoffmann-Jørgensen, using outer expectation, resolves this.<sup>[11](https://dept.stat.lsa.umich.edu/~moulib/emp-proc-notes-1-sep09.pdf)</sup> One workaround is to work with the supremum of expectations over finite subsets \( G \subseteq F \) of \( E \lVert P_n - P \rVert_G \), which is measurable because the maximum is over a finite set.<sup>[12](https://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/8notes.pdf)</sup>

## By the numbers: rates and constants

Dvoretzky, Kiefer, and Wolfowitz proved in a joint 1956 paper that \( P(\sup_x \lvert F(x) - \bar{F}_n(x) \rvert > a) \le C e^{-2 n a^2} \) for some absolute constant C; the optimal value C = 2 was established by Massart in 1990.<sup>[9](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/06/GlivenkoCantelli-summary.pdf)</sup> Dudley's notes record the 1956 result as holding for some constant C with \( 2 < C < \infty \), and Massart's theorem as the sharpened explicit version.<sup>[13](https://math.mit.edu/%7Ermd/465/edf-ks.pdf)</sup> In modern notation the two-sided DKW–Massart inequality reads

\[ P\Bigl( \sup_{x \in \mathbb{R}} \lvert \hat{F}_n(x) - F(x) \rvert > \varepsilon \Bigr) \le 2 e^{-2 n \varepsilon^2}, \quad \forall \varepsilon > 0, \]

and the one-sided form \( P(\sup_x (\hat{F}_n(x) - F(x)) > \varepsilon) \le e^{-2 n \varepsilon^2} \) holds for all \( \varepsilon > 0 \).<sup>[3](https://arxiv.org/html/2607.04387)</sup> There is a detail here: Massart proved the one-sided bound only for \( \varepsilon > \sqrt{\log 2 / (2n)} \), and Reeve later extended its validity to all \( \varepsilon > 0 \) using a martingale approach.<sup>[3](https://arxiv.org/html/2607.04387)</sup>

The asymptotic rate comes from the law of the iterated logarithm: \( \lVert F_n - F \rVert = O(n^{-1/2} (\log \log n)^{1/2}) \) with probability 1, so the supremum distance goes to zero only a little more slowly than \( n^{-1/2} \).<sup>[2](https://sites.stat.washington.edu/jaw/JAW-papers/NR/jaw-gaenssler.ess.83.pdf)</sup> For general classes, there is a universal constant L such that for any distribution and any class of sets with finite VC dimension, the uniform L1-type error is bounded by \( L \sqrt{\mathrm{vc}(F)/n} \), an explicit \( 1/\sqrt{n} \) rate whose dependence on class complexity enters only as a constant factor.<sup>[6](https://pillowmath.github.io/Expository%20Notes/VC-Dimension-and-Glivenko-Cantelli-Notes.pdf)</sup>

## The Kolmogorov–Smirnov test connection

The Kolmogorov–Smirnov statistic is \( KS = \sup_{x \in \mathbb{R}} \lvert \hat{F}_n(x) - F_0(x) \rvert \), a plug-in goodness-of-fit statistic that is small under the null precisely because of the Glivenko–Cantelli theorem; the Cramér–von Mises test is a related plug-in statistic.<sup>[8](https://www.stat.cmu.edu/~larry/=stat705/Lecture7.pdf)</sup> The NIST handbook defines the statistic as \( D = \max_i \bigl( F(Y_i) - \tfrac{i-1}{N},\ \tfrac{i}{N} - F(Y_i) \bigr) \), where F is the theoretical cumulative distribution being tested, and requires that F be continuous (no discrete distributions such as the binomial or Poisson) and fully specified, with location, scale, and shape parameters not estimated from the data.<sup>[14](https://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm)</sup> The term "Kolmogorov–Smirnov statistic" can refer to either \( \sup_x \lvert F(x) - \bar{F}_n(x) \rvert \) or \( \sqrt{n} \sup_x \lvert F(x) - \bar{F}_n(x) \rvert \).<sup>[9](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/06/GlivenkoCantelli-summary.pdf)</sup>

The DKW inequality is widely used in statistics for tests and for the construction of confidence bands for the CDF.<sup>[3](https://arxiv.org/html/2607.04387)</sup> Inverting it gives a 1−δ confidence band \( \hat{F}(t) \pm \sqrt{\log(2/\delta)/(2n)} \) around the empirical CDF, valid at all t simultaneously; this band is quite loose, and too wide away from the median. A tighter simulation-based alternative uses the 1−δ quantile \( \varepsilon_{\mathrm{sim}} \) of Kolmogorov–Smirnov statistics computed under a reference distribution, giving \( \hat{F}(t) \pm \varepsilon_{\mathrm{sim}} \); this is essentially the [Kolmogorov–Smirnov test](https://www.edgepedia.org/kolmogorov-smirnov-test).<sup>[4](https://stephenbates19.github.io/teaching/stat_theory_2024/notes/lecture10.pdf)</sup>

## Generalizations: classes, dependence, dimensions

Modern empirical process theory phrases the theorem for general function classes. A class F is a P-Glivenko–Cantelli class if \( \sup_{f \in F} \lvert P_n f - P f \rvert \to 0 \) almost surely as \( n \to \infty \); the classical theorem is the special case \( F = \{ \mathbf{1}[x \le t] : t \in \mathbb{R} \} \) with the stronger almost-sure conclusion.<sup>[5](https://statistique.cuso.ch/fileadmin/statistique/user_upload/3gcd.pdf)</sup><sup> • </sup><sup>[12](https://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/8notes.pdf)</sup> Not every class qualifies: for the class of indicators of finite sets and a continuous P, \( P f = 0 \) for every f in the class but \( \sup_{f} P_n f = 1 \) for all n, so convergence fails to be uniform because the class is too large.<sup>[12](https://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/8notes.pdf)</sup> Similarly, for multivariate data, uniform convergence holds on VC-classes of sets such as rectangular cells, but the class of all Borel sets is too rich for any finite-sample uniform convergence.<sup>[15](https://theorempath.com/topics/glivenko-cantelli-theorem)</sup>

**GC versus Donsker.** F is a P-Donsker class if \( \mathbb{G}_n \to \mathbb{G} \) in \( \ell_\infty(F) \), where the finite-dimensional distributions of \( \mathbb{G}_n \) converge to those of a centered [Gaussian process](https://www.edgepedia.org/gaussian-process) with a given covariance function.<sup>[5](https://statistique.cuso.ch/fileadmin/statistique/user_upload/3gcd.pdf)</sup> The distinction matters because a GC class guarantees a uniform law of large numbers, while a Donsker class additionally guarantees a functional central limit theorem at the \( \sqrt{n} \) scale; the modern theory measures class complexity by entropy numbers and VC dimensions to decide which classes satisfy each property.<sup>[11](https://dept.stat.lsa.umich.edu/~moulib/emp-proc-notes-1-sep09.pdf)</sup> A finer split separates universal GC classes, which are GC for all laws P on a measurable space, from uniform GC classes, for which \( P_n \) converges to P uniformly over the class almost surely uniformly in P.<sup>[16](https://link.springer.com/article/10.1007/BF01210321)</sup>

**Dependence.** Tucker generalized the classical theorem to strictly stationary sequences, with \( \lVert F_n - F_\omega \rVert \to 0 \) almost surely, where \( F_\omega \) is the common marginal law when the sequence is ergodic; for independent but not identically distributed observations, the average empirical measure \( P_n = n^{-1}(P_1 + \dots + P_n) \) replaces the common P.<sup>[2](https://sites.stat.washington.edu/jaw/JAW-papers/NR/jaw-gaenssler.ess.83.pdf)</sup> Recent work extends these results further: a 2025 preprint handles real-valued empirical function classes under α-mixing and β-mixing conditions, replacing specific mixing-rate requirements with a general criterion based on entropy conditions and VC classes.<sup>[17](https://arxiv.org/html/2502.20206v6)</sup> For graph-dependent data there is no graph-independent Glivenko–Cantelli rate; the rate is determined jointly by function-class complexity, graph geometry, and the decay of stochastic dependence with graph distance.<sup>[18](https://arxiv.org/html/2606.31936)</sup>

## How it compares with related limit theorems

The Glivenko–Cantelli theorem is a uniform version of the strong law of large numbers, applied to the particularly tractable indicator function \( \mathbf{1}(\cdot \le t) \); modern empirical process theory generalizes both the GC and Donsker results to arbitrary function classes through \( \lVert P_n - P \rVert_F = \sup_{f \in F} \lvert P_n f - P f \rvert \).<sup>[10](https://sites.stat.washington.edu/people/fanghan/teaching/STAT583/lec1.pdf)</sup> The three results sit at different levels: the strong law gives pointwise convergence, Glivenko–Cantelli upgrades it to uniform convergence of the empirical CDF, and the Donsker theorem describes the fluctuation scale, with \( \sqrt{n}(F - \bar{F}_n) \) converging weakly in the Skorokhod space \( D(\mathbb{R}) \) to \( B(F(x)) \), a [Brownian bridge](https://www.edgepedia.org/brownian-bridge) composition.<sup>[9](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/06/GlivenkoCantelli-summary.pdf)</sup>

## Role in learning theory and generalization bounds

The fundamental theorem of statistical learning makes uniform convergence exact for 0-1 loss: for a hypothesis class H, the following are equivalent: H has the uniform convergence property; any ERM rule is a successful agnostic PAC learner; H is agnostic PAC learnable; H is PAC learnable; any ERM rule is a successful PAC learner; and H has finite VC dimension.<sup>[7](https://weiningchen.github.io/notes/VC_dim.pdf)</sup>

The VC theorem reduces the question of uniform convergence to a combinatorial one about shatter coefficients; for a class of VC dimension d, Sauer's lemma gives \( P(\Delta(A) \ge t) \le 8 (n+1)^d e^{-n t^2 / 32} \), so with probability \( 1 - \delta \), \( \Delta(A) \le \sqrt{\tfrac{32}{n} \bigl[ d \log(n+1) + \log(8/\delta) \bigr]} \). If d is finite then \( \Delta(A) \to 0 \), a uniform law of large numbers, and converses show that if the VC dimension is infinite there exists a distribution for which no uniform law holds; roughly \( \Delta(A) \approx \sqrt{d \log n / n} \).<sup>[19](https://www.stat.cmu.edu/~siva/teaching/705/lec9.pdf)</sup> The shatter coefficients of left intervals are bounded by n+1, which immediately yields the classical Glivenko–Cantelli theorem as a corollary.<sup>[19](https://www.stat.cmu.edu/~siva/teaching/705/lec9.pdf)</sup>

For real-valued functions, Alon, Ben-David, Cesa-Bianchi, and Haussler showed that uniform Glivenko–Cantelli classes are exactly the classes for which means converge to expectations distribution-free uniformly, a property strongly tied to PAC learnability, and proved a characterization via a combinatorial quantity generalizing the VC dimension that recovers the Dudley, Giné, and Zinn characterization as a corollary.<sup>[20](https://dl.acm.org/doi/10.1145/263867.263927)</sup> Three routes to uniform convergence address the same phenomenon: VC dimension (the combinatorial route of classical PAC learning), [Rademacher complexity](https://www.edgepedia.org/rademacher-complexity) (the data-dependent route of margin-based bounds), and bracketing-entropy integrals (the empirical-process route of semiparametric statistics and [M-estimation](https://www.edgepedia.org/m-estimation)).<sup>[15](https://theorempath.com/topics/glivenko-cantelli-theorem)</sup>

## Recent developments

Several refinements have appeared recently. Bartl and Mendelson (2023) proved a variance-dependent DKW inequality with constants c and c′ for \( t \ge c \sqrt{\log \log n} \).<sup>[21](https://proceedings.mlr.press/v237/blanchard24a/blanchard24a.pdf)</sup> A COLT 2024 paper resolved an open problem of Cohen and Kontorovich by giving exact non-asymptotic instance-dependent bounds for the expected maximum deviation for product distributions on \( \{0,1\}^d \), including dimension-free results for infinite-dimensional p; the classical DKW rate \( \Delta_n(p) \lesssim \sqrt{\ln(d+1)/n} \) is optimal up to constants without further assumptions, being attained for a d-dimensional constant vector.<sup>[21](https://proceedings.mlr.press/v237/blanchard24a/blanchard24a.pdf)</sup> A 2025 follow-up showed that, once a learner is barred from exploiting pathological quirks of the infinite-dimensional setting, local Glivenko–Cantelli is essentially the largest learnable family, and the empirical mean estimator achieves the minimax risk decay rate over non-pathological distribution families.<sup>[22](https://raw.githubusercontent.com/mlresearch/v267/main/assets/cohen25b/cohen25b.pdf)</sup>

On the dependence side, a 2024 paper gives general upper bounds on empirical-process suprema under β- and ρ-mixing without assuming summability of mixing coefficients, covering both short- and long-range dependence, and uncovers a new phenomenon: even under long-range dependence, i.i.d.-level rates are attainable provided the underlying function class is complex enough.<sup>[23](https://arxiv.org/html/2401.08978v1)</sup> A 2025 concentration inequality for suprema of empirical processes with dependent data extends the classical i.i.d. results, with the bound depending on Talagrand's functional and a coupling correction governed by an "effective sample size" quantifying information loss due to dependence; applied to empirical risk minimization it yields a non-asymptotic oracle inequality in which i.i.d.-style prediction rates hold once sample size is replaced by effective sample size.<sup>[24](https://arxiv.org/html/2511.00597)</sup> In approximation theory, a 2025 [Annals of Statistics](https://www.edgepedia.org/annals-of-statistics) paper established a valid uniform Gaussian strong approximation at the optimal rate \( n^{-1/2} \log n \) for d = 2 empirical processes indexed by Lipschitz functions, a rate previously known only for univariate processes via the Hungarian construction of Komlós, Major, and Tusnády (1975), improving the rate from \( n^{-1/(2d)} \) to \( n^{-1/\max\{d,2\}} \) up to a polylogarithmic factor.<sup>[25](https://mdcattaneo.github.io/papers/Cattaneo-Yu_2025_AOS.pdf)</sup>

The reference literature has also moved: the second edition of van der Vaart and Wellner's *Weak Convergence and Empirical Processes* (2023) adds Glivenko–Cantelli preservation theorems, new bounds on expectations of suprema of empirical processes, new covering-number bounds, generic chaining, and definitive versions of concentration bounds, alongside applications to rates of convergence of estimators, M- and Z-estimators, the bootstrap, the functional delta-method, and semiparametric estimation.<sup>[26](https://link.springer.com/book/10.1007/978-3-031-29040-4)</sup>

## References

1. [The Glivenko–Cantelli Theorem, A. M. Shaikh, University of Chicago lecture notes](https://home.uchicago.edu/~amshaikh/webfiles/glivenko-cantelli_topics.pdf)
2. [Glivenko–Cantelli Theorems, Encyclopedia of Statistical Sciences (Gaenssler & Stute / Wellner)](https://sites.stat.washington.edu/jaw/JAW-papers/NR/jaw-gaenssler.ess.83.pdf)
3. [An Elementary Proof of the Dvoretzky–Kiefer–Wolfowitz–Massart Inequality, arXiv](https://arxiv.org/html/2607.04387)
4. [Lecture 10: DKW Inequality, Stephen Bates, 2024](https://stephenbates19.github.io/teaching/stat_theory_2024/notes/lecture10.pdf)
5. [Empirical Processes with Applications in Statistics, Ch. 3: Glivenko–Cantelli and Donsker Theorems](https://statistique.cuso.ch/fileadmin/statistique/user_upload/3gcd.pdf)
6. [VC Dimension and Glivenko–Cantelli Notes](https://pillowmath.github.io/Expository%20Notes/VC-Dimension-and-Glivenko-Cantelli-Notes.pdf)
7. [Introduction to the VC-Dimension, notes following Shalev-Shwartz & Ben-David](https://weiningchen.github.io/notes/VC_dim.pdf)
8. [Lecture Notes 7, Larry Wasserman, CMU Stat 705](https://www.stat.cmu.edu/~larry/=stat705/Lecture7.pdf)
9. [Glivenko–Cantelli summary, USC](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/06/GlivenkoCantelli-summary.pdf)
10. [Glivenko–Cantelli and Donsker, STAT 583 lecture notes, University of Washington](https://sites.stat.washington.edu/people/fanghan/teaching/STAT583/lec1.pdf)
11. [Empirical Processes: Notes 1, University of Michigan](https://dept.stat.lsa.umich.edu/~moulib/emp-proc-notes-1-sep09.pdf)
12. [Theoretical Statistics, Lecture 8, P. Bartlett, Berkeley Stat 210B](https://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/8notes.pdf)
13. [EDF and Kolmogorov–Smirnov test, R. M. Dudley, MIT 18.465 lecture notes](https://math.mit.edu/%7Ermd/465/edf-ks.pdf)
14. [1.3.5.16. Kolmogorov-Smirnov Goodness-of-Fit Test, NIST/SEMATECH e-Handbook](https://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm)
15. [Glivenko-Cantelli Theorem. Uniform Convergence of the Empirical CDF, TheoremPath](https://theorempath.com/topics/glivenko-cantelli-theorem)
16. [Uniform and universal Glivenko–Cantelli classes, Probability Theory and Related Fields](https://link.springer.com/article/10.1007/BF01210321)
17. [On the Glivenko-Cantelli theorem for real-valued empirical functions of stationary α-mixing and β-mixing sequences, arXiv](https://arxiv.org/html/2502.20206v6)
18. [Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes, arXiv](https://arxiv.org/html/2606.31936)
19. [Estimation of Statistical Functionals / VC theorem, CMU 10-705 lecture notes](https://www.stat.cmu.edu/~siva/teaching/705/lec9.pdf)
20. [Scale-sensitive dimensions, uniform convergence, and learnability, Alon et al., STOC 1993](https://dl.acm.org/doi/10.1145/263867.263927)
21. [Tight Bounds for Local Glivenko–Cantelli, Blanchard et al., COLT 2024](https://proceedings.mlr.press/v237/blanchard24a/blanchard24a.pdf)
22. [The Empirical Mean is Minimax Optimal for Local Glivenko-Cantelli, PMLR v267, 2025](https://raw.githubusercontent.com/mlresearch/v267/main/assets/cohen25b/cohen25b.pdf)
23. [Trade-off Between Dependence and Complexity for Nonparametric Learning, arXiv](https://arxiv.org/html/2401.08978v1)
24. [Concentration Inequalities for Suprema of Empirical Processes with Dependent Data via Generic Chaining, arXiv](https://arxiv.org/html/2511.00597)
25. [Strong approximations for empirical processes indexed by Lipschitz functions, Cattaneo & Yu, Annals of Statistics 2025](https://mdcattaneo.github.io/papers/Cattaneo-Yu_2025_AOS.pdf)
26. [Weak Convergence and Empirical Processes, 2nd ed., van der Vaart & Wellner, Springer 2023](https://link.springer.com/book/10.1007/978-3-031-29040-4)

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