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Weak convergence

In mathematics, weak convergence refers to a type of probabilistic or topological convergence that is weaker than stronger forms such as almost sure convergence or norm convergence, and it plays a central role in limit theorems and asymptotic analysis across probability theory and functional analysis.^[1]^[2] In probability theory, weak convergence describes the limiting behavior of sequences of probability measures or random variables, where a sequence of probability measures \{\mu_n\} on a metric space (S, \mathcal{S}) is said to converge weakly to a probability measure \mu (denoted \mu_n \Rightarrow \mu) if \int_S f \, d\mu_n \to \int_S f \, d\mu for every bounded continuous real-valued function f: S \to \mathbb{R}.^[1]^[3] Equivalently, for random variables X_n with distributions \mu_n, weak convergence to a random variable X with distribution \mu—also called convergence in distribution—occurs if the cumulative distribution functions F_n(x) \to F(x) at all continuity points x of F when S = \mathbb{R}.^[4] This mode of convergence depends solely on the marginal distributions and not on joint dependencies or sample paths, making it particularly useful for studying asymptotic properties without requiring pathwise control.^[1] Key properties include uniqueness of limits, preservation under continuous mappings (via the continuous mapping theorem), and characterization through the Portmanteau theorem, which equates weak convergence to conditions on measures of open, closed, or continuity sets.^[5] Its importance stems from foundational applications in central limit theorems, where normalized sums of independent random variables converge weakly to a normal distribution, enabling approximations in statistics and stochastic processes.^[1]^[3] Prohorov's theorem further links weak convergence to tightness and relative compactness of measure sequences in complete separable metric spaces, facilitating proofs of existence for limiting distributions.^[5] In functional analysis, weak convergence applies to sequences in normed linear spaces and is defined such that a sequence \{x_n\} in a space X converges weakly to x \in X (denoted x_n \rightharpoonup x) if \langle x_n, \mu \rangle \to \langle x, \mu \rangle for every continuous linear functional \mu \in X^*, the topological dual of X.^[2] Unlike strong (norm) convergence, which requires \|x_n - x\| \to 0, weak convergence implies only boundedness of the sequence and uniqueness of the limit, but it does not preserve norms in infinite-dimensional spaces—strong convergence implies weak, but the converse fails.^[2] This weaker topology is essential for compactness results like Alaoglu's theorem in Banach spaces and for variational methods in partial differential equations, where weak limits preserve integrals against test functions.^[2]

In Probability and Measure Theory

Definition and Basic Concepts

Weak convergence is a fundamental concept in probability theory that describes the limiting behavior of sequences of probability distributions in a topological sense, particularly on metric spaces. It generalizes the idea of convergence in distribution for random variables and is essential for limit theorems in stochastic processes.^[6] Consider a metric space (X, d). A sequence of probability measures \{\mu_n\}_{n=1}^\infty on the Borel \sigma-algebra of X is said to converge weakly to a probability measure \mu if, for every bounded continuous function f: X \to \mathbb{R},

\int_X f \, d\mu_n \to \int_X f \, d\mu

as n \to \infty.^[6] This definition arises from viewing probability measures as elements of the dual space of C_b(X), the Banach space of bounded continuous real-valued functions on X equipped with the supremum norm.^[6] The weak topology on the space \mathcal{P}(X) of probability measures is the coarsest topology making these integral maps continuous.^[6] Results are often developed for Polish spaces—separable complete metric spaces—where the Borel \sigma-algebra supports rich structure and compactness arguments apply effectively.^[6] A key prerequisite for weak convergence is the notion of tightness, which ensures that mass does not escape to infinity. A family \{\mu_i\} of probability measures on X is tight if, for every \epsilon > 0, there exists a compact subset K \subset X such that \mu_i(K) \geq 1 - \epsilon for all i.^[7] Tightness is necessary for relative compactness in \mathcal{P}(X) under the weak topology, meaning every sequence in a tight family has a weakly convergent subsequence.^[7] The Prokhorov metric provides a way to metrize this weak topology on separable metric spaces. For probability measures \mu, \nu \in \mathcal{P}(X), it is defined by

d(\mu, \nu) = \inf\left\{ \epsilon > 0 : \mu(A) \leq \nu(A^\epsilon) + \epsilon \ \forall \text{ Borel sets } A \subset X, \text{ and vice versa} \right\},

where A^\epsilon = \{ x \in X : d(x, A) < \epsilon \} denotes the open \epsilon-neighborhood of A.^[7] A sequence \{\mu_n\} converges weakly to \mu if and only if d(\mu_n, \mu) \to 0.^[7] This metric was introduced by Prokhorov in his 1956 paper on convergence in metric spaces.^[7]

Portmanteau Theorem and Equivalences

The Portmanteau theorem provides a fundamental characterization of weak convergence of probability measures on a separable metric space (S, \mathcal{S}), where \{\mu_n\} is a sequence of probability measures converging weakly to a probability measure \mu.^[8] Specifically, the following conditions are equivalent: (i) \int f \, d\mu_n \to \int f \, d\mu for every bounded continuous function f: S \to \mathbb{R}; (ii) \limsup_n \mu_n(F) \leq \mu(F) for every closed set F \in \mathcal{S}; (iii) \liminf_n \mu_n(G) \geq \mu(G) for every open set G \in \mathcal{S}; and (iv) \mu_n(A) \to \mu(A) for every Borel set A \in \mathcal{S} such that \mu(\partial A) = 0, where \partial A denotes the boundary of A.^[8] These conditions offer practical tools for verifying weak convergence without directly computing integrals over all continuous functions, as the portmanteau inequalities for closed and open sets capture the topological behavior of measures.^[9] An additional equivalence arises from convergence against indicator functions: \int 1_A \, d\mu_n \to \int 1_A \, d\mu for every continuity set A (i.e., \mu(\partial A) = 0), which aligns directly with condition (iv) of the Portmanteau theorem and extends the integral condition to a broader class of bounded measurable functions under the continuity assumption.^[8] On the real line \mathbb{R}, weak convergence of distribution functions F_n to F is equivalently metrized by convergence in the Lévy metric \lambda, defined as \lambda(F_n, F) = \inf \{ \epsilon > 0 : F_n(x - \epsilon) - \epsilon \leq F(x) \leq F_n(x + \epsilon) + \epsilon \ \forall x \in \mathbb{R} \}, providing a quantitative measure of convergence for one-dimensional cases.^[9] A concrete illustration of these equivalences appears in the Glivenko-Cantelli theorem, which establishes the almost sure uniform convergence of the empirical cumulative distribution function (CDF) \hat{F}_n(x) = n^{-1} \sum_{i=1}^n 1_{\{X_i \leq x\}} to the true CDF F for i.i.d. samples X_1, \dots, X_n from a distribution on \mathbb{R}.^[10] For the uniform distribution on [0,1], where F(x) = x for x \in [0,1], the empirical measure \mu_n = n^{-1} \sum_{i=1}^n \delta_{X_i} converges weakly to the Lebesgue measure on [0,1], as verified by the Portmanteau condition (iv): for continuity sets like intervals (a,b) with $0 < a < b < 1, \mu_n((a,b)) \to b - a almost surely, and the uniform CDF convergence implies the Lévy metric condition \lambda(\hat{F}_n, F) \to 0.^[10] This example highlights how empirical measures approximate true distributions, with the supremum norm \sup_x |\hat{F}_n(x) - F(x)| \to 0 ensuring all Portmanteau conditions hold.^[10] To address weak convergence in non-metric spaces, such as locally compact Hausdorff spaces, the Portmanteau theorem extends via vague convergence of finite measures, defined by convergence of integrals over continuous functions with compact support: \int f \, d\mu_n \to \int f \, d\mu for all such f.^[11] Analogous portmanteau conditions include \limsup_n \mu_n(K) \leq \mu(K) for compact K and \liminf_n \mu_n(U) \geq \mu(U) for open U with compact closure, enabling characterizations in settings beyond metric topologies while preserving the theorem's utility for verification.

Skorokhod Representation

The Skorokhod representation theorem provides a constructive realization of weak convergence in terms of almost sure convergence of random variables. Specifically, if a sequence of probability measures \{\mu_n\}_{n=1}^\infty on the Borel \sigma-algebra of a Polish space (S, d) converges weakly to a probability measure \mu, then there exists a probability space (\Omega, \mathcal{F}, P) and S-valued random variables X_n: \Omega \to S and X: \Omega \to S such that the law of X_n is \mu_n, the law of X is \mu, and X_n \to X almost surely with respect to P.^[12] A proof sketch begins with the case S = \mathbb{R}^d. Let U be a random vector uniformly distributed on [0,1]^d, independent of the sequence. Define X_n = F_n^{-1}(U), where F_n^{-1} is the generalized inverse (quantile function) of the cumulative distribution function corresponding to \mu_n, and similarly X = F^{-1}(U) for the limit \mu. Weak convergence \mu_n \to \mu implies that the quantile functions F_n^{-1}(u) \to F^{-1}(u) for Lebesgue-almost every u \in [0,1]^d, since points of discontinuity of F^{-1} form a set of Lebesgue measure zero. Thus, X_n \to X almost surely.^[13] For general Polish spaces, the construction extends by leveraging the separability and completeness of S. One approach embeds S into the unit interval via a measurable bijection or uses the Prokhorov metric to find a common probability space where the random variables can be defined via measurable selections that preserve the laws and ensure almost sure convergence, often relying on the metrizability of weak convergence on such spaces. The representation is not unique, as multiple couplings of the measures can achieve almost sure convergence; for instance, one may modify the construction on sets of measure zero without altering the laws. Extensions beyond Polish spaces exist but require additional conditions. For separable metric spaces lacking completeness, the theorem holds via refinements that construct the variables without relying on completeness directly. For non-separable limit measures, versions demand convergence in stronger metrics like the Wasserstein distance or separability of the support to ensure the existence of such random variables.^[14] The theorem originated in a 1956 paper by A. V. Skorokhod, where it was introduced to facilitate proofs of limit theorems for stochastic processes. It finds applications in invariance principles, such as embedding partial sums into Brownian motion, by enabling almost sure approximations that simplify verification of weak convergence criteria like those in the Portmanteau theorem.^[12]

In Functional Analysis

Weak Topology on Normed Spaces

In a normed vector space X with dual space X^*, the weak topology, denoted \sigma(X, X^*), is the coarsest topology on X such that every continuous linear functional f \in X^* is continuous. This topology is generated by the family of seminorms p_f(x) = |f(x)| for all f \in X^*, or equivalently, by the subbasis consisting of sets of the form \{x \in X : |f(x) - f(x_0)| < \varepsilon\} where f \in X^*, x_0 \in X, and \varepsilon > 0. The open sets in this topology, known as cylinder sets, are finite intersections of such subbasis elements, providing a coordinate-like structure aligned with the action of the dual space.^[15] The weak topology is strictly coarser than the norm topology on X whenever X is infinite-dimensional, meaning every norm-open set contains a weak-open set, but the converse fails.^[16] For instance, consider X = \ell^p for $1 < p < \infty with dual X^* = \ell^q where \frac{1}{p} + \frac{1}{q} = 1; the standard basis vectors e_n = (0, \dots, 0, 1, 0, \dots) (with 1 in the nth position) converge to 0 in the weak topology since f(e_n) = f_n \to 0 for every f = (f_k) \in \ell^q (as q < \infty implies f_k \to 0), but \|e_n\| = 1 \not\to 0 in the norm topology.^[17] The Hahn-Banach separation theorem ensures that the weak topology is Hausdorff: for distinct x, y \in X, there exists f \in X^* with f(x) \neq f(y), so the subbasis sets separate points.^[16] This property relies on the theorem's extension of linear functionals while preserving continuity. The weak topology's development is tied to early functional analysis; notably, Alaoglu's 1940 theorem establishes weak* compactness of the closed unit ball in X^* under the weak* topology \sigma(X^*, X), highlighting compactness features absent in the weak topology on non-reflexive spaces (details in subsequent sections).

Weak Convergence of Sequences

In a normed linear space X, a sequence (x_n) converges weakly to an element x \in X if f(x_n) \to f(x) for every continuous linear functional f \in X^*.^[18] This notion of convergence aligns with the weak topology on X, where the dual separates points, ensuring that weak limits, when they exist, are unique.^[19] A key property of weakly convergent sequences is that they are norm bounded. To see this, suppose (x_n) converges weakly to x. For each fixed f \in X^*, the sequence f(x_n) is convergent and thus bounded. The uniform boundedness principle, also known as the Banach–Steinhaus theorem, implies that the family of point evaluation maps at the x_n (viewed as elements inducing functionals on X^*) is uniformly bounded on the unit ball of X^*, yielding \sup_n \|x_n\| < \infty.^[20] This boundedness is essential for detecting weak limits, as unbounded sequences cannot converge weakly. The Eberlein–Šmulian theorem provides a sequential criterion for weak compactness in Banach spaces: a subset A of a Banach space X is relatively weakly compact if and only if every sequence in A admits a weakly convergent subsequence. This result, originally established by Eberlein for general sets and extended by Šmulian to convex sets, bridges topological compactness with sequential behavior in the weak topology. A significant consequence concerns reflexivity: a Banach space X is reflexive if and only if its closed unit ball is weakly compact, which, by the Eberlein–Šmulian theorem, is equivalent to every bounded sequence in X having a weakly convergent subsequence.^[21] This characterization highlights how weak sequential compactness detects reflexive structure, distinguishing spaces like Hilbert spaces from non-reflexive ones such as c_0. In Hilbert spaces, weak convergence admits a concrete interpretation via the Riesz representation theorem, which identifies the dual with the space itself through inner products. Thus, x_n \rightharpoonup x if and only if \langle x_n, y \rangle \to \langle x, y \rangle for all y in the space, corresponding to convergence under orthogonal projections onto one-dimensional subspaces spanned by each y. For a concrete example, consider the Hilbert space \ell^2(\mathbb{N}) with the standard orthonormal basis (e_j)_{j \in \mathbb{N}}, where e_j(k) = \delta_{jk}. This sequence converges weakly to 0, since for any g = (g_k) \in \ell^2(\mathbb{N}), \langle g, e_j \rangle = g_j \to 0 as j \to \infty (by the square-summability of g). However, \|e_j\| = 1 for all j, so the norms do not converge to 0, illustrating that weak convergence does not imply strong (norm) convergence in infinite-dimensional settings.^[22]

Reflexivity and Weak Compactness

In the context of weak convergence in Banach spaces, reflexivity plays a pivotal role in ensuring desirable compactness properties. A Banach space X is reflexive if the canonical embedding J: X \to X^{**} is surjective, meaning X = J(X), where X^{**} is the bidual. This property implies that the weak topology \sigma(X, X^*) on bounded sets of X coincides with the weak* topology \sigma(X^{**}, X^*) restricted to X, facilitating the study of weak limits and compactness.^[23] James' theorem provides a characterization of reflexivity in terms of norm attainment: A Banach space X is reflexive if and only if every continuous linear functional f \in X^* attains its norm on the closed unit ball B_X = \{x \in X : \|x\| \leq 1\}, that is, there exists x \in B_X such that |f(x)| = \|f\|. This result highlights how failure of reflexivity manifests in the non-attainment of suprema by functionals on bounded sets, linking directly to the absence of weak compactness in non-reflexive spaces.^[23] A cornerstone result for weak compactness is Alaoglu's theorem, which states that the closed unit ball B_{X^*} = \{f \in X^* : \|f\| \leq 1\} in the dual space X^* is compact in the weak* topology \sigma(X^*, X). In reflexive spaces, where X^{**} = X, the weak topology \sigma(X^*, X^{**}) on X^* coincides with the weak* topology \sigma(X^*, X), so B_{X^*} is also compact in the weak topology. Consequently, the unit ball B_X of a reflexive space X is weakly compact, as it is the polar of B_{X^*} and inherits compactness via the bipolar theorem. This equivalence underscores reflexivity as the condition for the unit ball to be weakly compact.^[24] In reflexive Banach spaces, the Krein-Milman theorem further elucidates the structure of the weakly compact unit ball: B_X equals the closed convex hull of its extreme points, where an extreme point of B_X is a point x \in B_X that cannot be written as a nontrivial convex combination of distinct points in B_X. This representation emphasizes the role of extreme points in approximating elements via weak limits of convex combinations. A classic counterexample illustrating the consequences of non-reflexivity is the Banach space c_0 of real sequences converging to zero, equipped with the supremum norm \|x\|_\infty = \sup_n |x_n|. The space c_0 is not reflexive, as its bidual is \ell^\infty, the space of all bounded sequences, and c_0 \neq \ell^\infty. In this setting, weak convergence in c_0, defined via the weak topology \sigma(c_0, \ell^1), differs from weak* convergence when c_0 is embedded into \ell^\infty via the weak* topology \sigma(\ell^\infty, \ell^1), since the larger dual \ell^\infty allows limits outside c_0 that are incompatible with weak limits in c_0. For instance, nets in the unit ball of c_0 can converge weak* in \ell^\infty to elements not in c_0, demonstrating that the unit ball of c_0 fails to be weakly compact.

Comparisons and Relations

Strong versus Weak Convergence

In normed linear spaces, strong convergence of a sequence (x_n) to a limit x is defined by \|x_n - x\| \to 0 as n \to \infty, whereas weak convergence requires that f(x_n) \to f(x) for every continuous linear functional f on the space. Strong convergence always implies weak convergence, since the norm can be recovered from the dual pairings via \|x\| = \sup_{\|f\| \leq 1} |f(x)|, but the reverse implication fails in infinite-dimensional settings.^[19] Weakly convergent sequences are necessarily bounded, by the uniform boundedness principle applied to the pointwise bounded family of functionals n \mapsto f(x_n) for each f in the dual space. However, boundedness alone does not ensure norm convergence, as the weak topology is coarser than the norm topology in infinite dimensions. In finite-dimensional normed spaces, all linear topologies coincide, so strong and weak convergence are equivalent.^[19] A canonical example illustrating the distinction occurs in the Hilbert space \ell^2, where the standard orthonormal basis vectors e_n (with 1 in the nth coordinate and 0 elsewhere) converge weakly but not strongly to 0. Indeed, \langle e_n, y \rangle = y_n \to 0 for any y = (y_k) \in \ell^2 (since \sum |y_k|^2 < \infty implies y_k \to 0), yet \|e_n\| = 1 \not\to 0.^[19] In the reflexive spaces \ell^p for $1 < p < \infty, weak convergence of (x_n) to x combined with \|x_n\| \to \|x\| implies strong (norm) convergence, as the limit satisfies the lower semicontinuity of the norm in the weak topology. This condition highlights when the two notions align beyond finite dimensions.

Metric Entropies and Tightness

In the context of weak convergence of probability measures on metric spaces, tightness plays a crucial role in establishing relative compactness. A family of probability measures \{\mu_n\} on a complete separable metric space (S, d) is tight if for every \varepsilon > 0, there exists a compact set K \subset S such that \mu_n(S \setminus K) < \varepsilon for all n. By Prokhorov's theorem, in Polish spaces, tightness is equivalent to relative compactness in the weak topology, meaning every sequence in the family has a weakly convergent subsequence.^[8] For measures on \mathbb{R}^d, tightness admits a refinement in terms of moments. Specifically, the family \{\mu_n\} is tight if and only if there exists p > 0 such that \{|X_n|^p\} is uniformly integrable, where X_n are random variables with laws \mu_n, or equivalently, \sup_n \mathbb{E}[|X_n|^p] < \infty. This condition ensures that the measures do not "escape to infinity" and facilitates weak convergence criteria in finite-dimensional spaces.^[8] In general metric spaces, Prokhorov's theorem extends tightness to characterize weak compactness without relying on finite-dimensional projections, applying directly to abstract Polish spaces and enabling weak convergence analysis for measures on function spaces or manifolds.^[8] Metric entropy provides a quantitative tool for verifying tightness, particularly for measures supported on infinite-dimensional spaces. The covering number N(\varepsilon, S, d) counts the minimal number of \varepsilon-balls needed to cover S. For Gaussian measures, Dudley's entropy integral bounds the expected supremum of the process: if \{X_t\}_{t \in T} is a centered Gaussian process on a metric space (T, \rho) with \rho(s,t) = \sqrt{\mathbb{E}(X_s - X_t)^2}, then \mathbb{E} \sup_{t \in T} |X_t| \leq C \int_0^\infty \sqrt{\log N(\varepsilon, T, \rho)} \, d\varepsilon < \infty, implying tightness of the induced measure on the path space. This integral condition ensures the sample paths remain controlled, bridging entropy to weak convergence of Gaussian processes. For stochastic processes on [0,1], the Kolmogorov-Chentsov theorem offers a moment-based criterion for tightness in the uniform topology. If \{X_n(t)\}_{t \in [0,1]} satisfies \mathbb{E} |X_n(t) - X_n(s)|^\alpha \leq C |t - s|^{1 + \beta} for some \alpha, \beta > 0 and C < \infty independent of n, then the family is tight in C[0,1] and the paths are Hölder continuous with exponent \gamma < \beta / \alpha almost surely. This theorem is pivotal for proving weak convergence of processes via finite-dimensional distributions plus tightness.^[8] Recent extensions address entropy bounds for non-Gaussian cases, adapting chaining arguments to sub-exponential or bounded processes. For instance, in empirical optimal transport under estimated costs, distributional limits and weak convergence are established for the transport costs when measures and costs converge weakly, enabling applications in statistical inference and machine learning.^[25] Such results provide controls for high-dimensional measures in statistical applications.

Applications in Stochastic Processes

Weak convergence plays a pivotal role in establishing functional central limit theorems for stochastic processes, most notably through Donsker's invariance principle, which demonstrates the convergence of scaled random walks to Brownian motion. Specifically, for a sequence of i.i.d. random variables \{\xi_i\} with mean zero and finite variance \sigma^2 > 0, the process X_n(t) = \frac{1}{\sigma \sqrt{n}} S_{\lfloor nt \rfloor}, where S_k = \sum_{i=1}^k \xi_i, converges weakly in the Skorokhod space D[0,1] to a standard Brownian motion W(t) as n \to \infty. This result, often termed the functional central limit theorem, extends the classical central limit theorem to the entire path of the process and relies on tightness conditions in the Skorokhod topology to handle discontinuities. The principle has broad implications for approximating discrete-time processes by continuous diffusions, enabling the analysis of queueing systems and diffusion approximations in operations research. In the theory of large deviations, weak convergence underpins key representations for rate functions, particularly through Varadhan's integral lemma, which links the logarithmic asymptotics of moment-generating functions to infima over function spaces. For a sequence of probability measures P_n satisfying a large deviation principle with speed n and good rate function I, the lemma states that if \sup f < \infty, then

\lim_{n \to \infty} \frac{1}{n} \log \int e^{n f} \, dP_n = \sup_{x} \left( f(x) - I(x) \right),

provided the supremum is attained and I is lower semicontinuous. This representation facilitates the derivation of rate functions for empirical measures and Markov processes, connecting weak convergence limits to exponential tail behaviors in stochastic systems like interacting particle models. Varadhan's result, integral to the weak convergence approach in large deviations, has been instrumental in proving principles for diffusion processes and Sanov-type theorems for empirical distributions. Empirical processes provide another arena where weak convergence manifests through uniform Glivenko-Cantelli theorems, ensuring almost sure convergence of sup-norms that aligns with weak limits in the space \ell^\infty. For i.i.d. observations from a distribution P on [0,1], the empirical process \sqrt{n} (\mathbb{P}_n - P) in \ell^\infty([0,1]) converges weakly to a Brownian bridge if the class of functions is a Donsker class, implying the uniform Glivenko-Cantelli property where \sup_{t \in [0,1]} |F_n(t) - F(t)| \to 0 almost surely, with F_n the empirical CDF.^[26] This convergence enables bootstrap approximations and confidence bands for statistical inference, extending classical results to Vapnik-Chervonenkis classes and handling high-dimensional settings via entropy bounds.^[26] Recent advancements in reinforcement learning leverage weak convergence to analyze policy gradient methods, particularly in continuous-time Markov decision processes where discrete approximations converge to limiting dynamics. For instance, in finite-horizon continuous-time RL, policy gradient updates derived from Euler-Maruyama discretizations of stochastic differential equations exhibit weak convergence to the true continuous-time gradient as the time step vanishes, ensuring asymptotic optimality under smoothness assumptions on the value function. This framework addresses exploration challenges in policy optimization by quantifying the bias in discrete implementations, with applications to mean-field games and multi-agent settings where weak limits preserve stationarity in the policy iteration. Such results, building on Skorokhod representations for pathwise coupling, enhance sample efficiency in high-dimensional control problems like robotics and finance.^[27]

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[PDF] Empirical Optimal Transport under Estimated Costs - arXiv
Jan 4, 2023 · ) represents a tight (possibly non-Gaussian) random variable on R2K+K2 . ... it follows by Dudley's entropy integral (see, e.g.,Wainwright, 2019, Chapter 5) that.<|control11|><|separator|>
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Weak Convergence and Empirical Processes - SpringerLink
Weak Convergence and Empirical Processes. With Applications to Statistics ... Stochastic Convergence. Front Matter. Pages 1-1. Download chapter PDF · Introduction.