Donsker's Invariance Principle

• Donsker's invariance principle is a fundamental result linking discrete random walks to continuous Brownian motion through weak convergence in rescaled processes.
• It establishes convergence in function spaces like C([0,1]) and D([0,1]), underpinning techniques in stochastic processes and statistical inference.
• Modern extensions address dependent data, sub-linear expectations, and thresholded weak convergence, broadening its applicability in advanced probability theory.

Donsker's invariance principle, also known as the functional central limit theorem (CLT), provides a probabilistic framework in which rescaled partial-sum or empirical processes converge in distribution to Brownian motion or related Gaussian processes. This principle underpins much of modern probability theory, stochastic processes, and statistical inference, serving as the foundational result linking discrete random walks to continuous-time diffusive phenomena. The scope of Donsker-type results now spans discrete and continuous time, i.i.d. and dependent structures, regular and irregular function spaces, and various classes of limit processes, with recent advances illuminating the boundaries of classic tightness and the phenomenon of thresholded weak convergence.

1. Classical Donsker's Principle and Its Scope

The classical Donsker theorem asserts that for centered i.i.d. variables $(X_i)_{i\ge1}$ with $\mathbb{E}[X_1]=0$ and $\operatorname{Var}(X_1)=\sigma^2<\infty$, the interpolated process

$$S_n(t) = \frac{1}{\sigma\sqrt{n}} \left[ \sum_{i=1}^{\lfloor nt \rfloor} X_i + (nt - \lfloor nt \rfloor) X_{\lfloor nt \rfloor+1} \right], \quad t\in[0,1],$$

converges in $C([0,1])$ to standard Brownian motion $W$. Analogous statements hold in the Skorokhod $D([0,1])$ topology and for the centered, rescaled empirical process (the empirical CDF normalized by $\sqrt{n}$), which converges to Brownian bridge. The theorem has been generalized to the multivariate setting, random fields, and functional data, always requiring tightness (asymptotic equicontinuity) and finite-dimensional convergence arguments.

Key variants of the classical result include:

• Empirical process version: Convergence in $\ell^\infty(\mathbb{R})$ to Gaussian processes with explicit covariance (e.g., $F(x\wedge y)-F(x)F(y)$ for empirical processes) (Su et al., 2017).
• Pathwise quantification: Wasserstein-1 distance ($W_1$) rates and explicit control of convergence in probability metrics on path space (Coutin et al., 2019).

2. Extensions: Dependent Data, Incomplete Sampling, and Sub-linear Expectations

Donsker-type theorems extend to processes with dependence and adversarial data deletions. For stationary martingale differences and ergodic dynamical systems, weak invariance principles can be established in Banach and Besov spaces, quantifying convergence and optimality in terms of modulus-of-smoothness and moment conditions (Giraudo et al., 2017). In "incomplete" or "deleting-item" processes—where an asymptotically negligible fraction of data is removed—functional CLTs remain valid under mild regularity assumptions; the functional limit persists as Brownian motion or bridge even for arbitrary deletion patterns, enriching the structure of weak convergence in Skorokhod spaces (Liu, 2019).

Under sub-linear expectation (a framework modeling model uncertainty or Knightian uncertainty), Donsker's principle holds with the limiting process as $G$-Brownian motion, characterized by volatility uncertainty and nonlinear capacity. Tightness is established via maximal inequalities in the sub-linear setting, and applications include small deviation asymptotics and nonlinear laws of the iterated logarithm (Zhang, 2015).

3. Donsker Principles for Stochastic Processes: Diffusions, Lévy Processes, Moving Averages

The field of stochastic processes has seen important Donsker-type invariance results:

• Diffusion processes: For multi-dimensional elliptic diffusions with periodic coefficients, the empirical occupation measure's rescaled deviations (empirical process) satisfy a Donsker theorem in $\ell^\infty(\mathcal{F})$, for classes $\mathcal{F}$ of smooth functions (specifically, bounded subsets of Besov spaces $B^s_{pq}$ with $s > \max\{d/2, d/p\} - 1$). The generator yields a "two-degree smoothing" effect, enlarging the class of Donsker-smooth functions compared to the i.i.d. case, with applications to sharp Wasserstein-1 distance bounds between empirical and invariant measures (Deo, 2022).
• Lévy processes: Both high-frequency and low-frequency sampling regimes have associated Donsker results for empirical and deconvolution functionals of Lévy measures. These demonstrate convergence to Gaussian processes (generalized Brownian bridges) with covariance governed by integral operators determined by the Lévy triplet, enabling uniform confidence bands and Kolmogorov–Smirnov-type inference for the Lévy jump measure (Nickl et al., 2013, Nickl et al., 2012).
• Self-normalization: For self-normalized sums $Y_{n,p}(t) = S_n(t)/V_{n,p}$, Donsker-type convergence to Brownian motion holds if and only if $p = \alpha = 2$; in other cases, the limiting process is degenerate or tightness fails (Basak et al., 2010, Parczewski, 2016). Sphere-based geometric proofs provide a unified framework for both classical and self-normalized invariance principles (Parczewski, 2016).
• Moving averages and memory: For finite-order moving averages with regularly varying memory, the normalized partial sum process converges to fractional integrals of fractional Brownian motion, extending the classical invariance principle to long-range dependent and heavy-tailed memory settings (Arkashov, 2022).

4. Functional Spaces, Regularity, and the Edge of Donsker's Principle

Donsker’s theorem holds in the space $C([0,1])$ or, with weaker topologies, in spaces of càdlàg functions. There is, however, a critical boundary in function space regularity. Brownian motion almost surely has modulus of continuity

$\rho_2(h) = \sqrt{h\log(e/h)},$

so the largest space supporting uniform tightness (and hence functional CLT) is the associated Hölder-Zygmund space $C^{\rho}$ with $\rho \gg \rho_2$. At the critical modulus $\rho = \rho_2$, tightness fails: the law-of-the-iterated-logarithm-scale fluctuations preclude classical weak convergence. Nevertheless, convergence of upper tails of supremum seminorms $\|\cdot\|_{\rho_2}$—termed "thresholded weak convergence"—can be established, meaning for large thresholds $t>\tau$, $$P(\|\widetilde S_n\|_{\rho_2}>t) \to P(\sigma\|B\|_{\rho_2}>t)$$, with critical value $\tau$ determined by the sub-Gaussian moment generating function (Köhne et al., 5 Jun 2025).

In Besov and related function spaces, tightness and weak invariance require explicit control of modulus-of-smoothness and higher moments, with sharp necessary and sufficient conditions for Donsker property formulated in terms of tail behavior of the innovations and conditional variances (Giraudo et al., 2017, Coutin et al., 2019). For diffusion empirical processes, two-degree smoothing arising from the elliptic generator allows Donsker theorems in strictly larger classes of smoothness than in i.i.d. settings (Deo, 2022).

5. Contemporary Extensions: Statistical Methodology and Multiscale Inference

Modern applications utilize Donsker-type theorems beyond foundational probability, notably in statistical methodology for inference:

• Multiscale scan statistics: Critical values for detection tasks in nonparametric regression and changepoint analysis require calibration under non-Gaussian and possibly nonstationary noise. Classical approach using additive penalties entails infeasibility outside Gaussian settings. New work establishes that multiplicative weighting and thresholded convergence in critical Hölder seminorms suffice for feasible and asymptotically efficient detection, with bootstrap schemes extending validity to time-varying and dependent error structures (Köhne et al., 5 Jun 2025).
• Goodness-of-fit and hypothesis testing: Donsker-type functional CLTs for log-likelihood processes indexed by parameter sets allow Gaussian (and Brownian bridge) limits for suprema over families of alternatives, essential for nonparametric and semiparametric global inference. These are achieved under entropy and Hellinger distance regularity controls for general semimartingale settings (Su et al., 2017).
• Almost sure invariance principles (ASIP): For random dynamical systems with spectral gap and mixing, Donsker’s theorem can be upgraded to almost-sure couplings, i.e., coupling the original process with a Brownian motion with small error, from which all classical probabilistic limit theorems can be inferred (Atnip, 2017).

6. Summary Table: Classical vs. Modern Donsker-Type Results

Setting	Limiting Process	Functional Space	Critical Tightness Condition
i.i.d. partial sums	Brownian motion	$C([0,1])$, $D([0,1])$	$\mathbb{E}|X_1|^2<\infty$
Empirical process	Brownian bridge	$\ell^\infty(\mathbb{R})$	VC class, entropy bounds
Martingale differences	Brownian motion	Besov $B^s_{p,\alpha}$	Tail + conditional moment req.
Diffusions (occupation)	Gaussian process	$\ell^\infty(\mathcal{F})$	$s>\max\{d/2, d/p\} - 1$
Lévy measure estimators	Gen. Brownian bridge	$\ell^\infty(\{|t|\ge\epsilon\})$	Fourier-analytic smoothness+mom.
Critical Hölder seminorm	BM $\|\cdot\|_{\rho_2}$, thresholded	$C^{\rho_2}$	Thresholded weak convergence

7. Impact and Limitations

The Donsker principle, with its myriad extensions, is central to the analysis of random processes and the construction of asymptotically exact inferential methods. Its validity in function spaces critically depends on the regularity of the process and the topology selected, with sharp boundaries determined by the modulus of continuity of the underlying limit process. In many settings, the presence of smoothing operators, dependence, self-normalization, or heavy tails can either enlarge the Donsker class (as in diffusions), or severely restrict it (as in heavy-tailed sums), with threshold phenomena at critical regularity. Recent theoretical advances at the edge of Donsker’s theorem, notably in thresholded weak convergence and feasible multiscale inference, have enabled a new generation of robust statistical methodology valid under minimal assumptions and in the presence of complex dependencies.

References: (Deo, 2022, Giraudo et al., 2017, Köhne et al., 5 Jun 2025, Nickl et al., 2013, Nickl et al., 2012, Coutin et al., 2019, Su et al., 2017, Zhang, 2015, Atnip, 2017, Parczewski, 2016, Basak et al., 2010, Arkashov, 2022, Liu, 2019)