Factor-Projected Empirical Process

• Factor-projected empirical process is a generalization of the classical empirical process that projects data onto smoother, lower-dimensional spaces using function classes like Hölder or Lipschitz.
• It establishes weak convergence and central limit theorems in settings with latent structures, strong dependencies, and high-dimensional or ill-conditioned data.
• Practical applications include model checking in dynamical systems, semiparametric factor models, and variance reduction methods for statistical inference.

A factor-projected empirical process is a generalization of the classical empirical process, specifically constructed to analyze settings where the data exhibit latent structure or dependence that can be modeled via factors, projections, or smooth functional classes. The central theme is that, rather than conducting all probabilistic or inferential arguments directly with respect to potentially nonsmooth or high-variance indexing functions (notably, indicator functions), one "projects" the empirical process onto a smoother or lower-dimensional factor space, often using function classes with stronger regularity or using data-driven functional projections. This approach enables rigorous limit theory and quantitative statistical analysis in high-dimensional, dependent, or otherwise ill-conditioned data settings such as dynamical systems, time series with Markovian structure, matrix-variate observations, or "semiparametric" factor models.

1. Framework, Motivation, and Main Construction

Let $(X_i)_{i\geq 0}$ be a $\mathbb{R}^d$-valued stationary (or locally stationary) process, and let $U_n(t)$ be the standard empirical process,

$$U_n(t_1,\ldots,t_d) = \frac{1}{\sqrt{n}} \sum_{i=1}^n \left[ \prod_{j=1}^d 1_{(-\infty, t_j]}(X_{i,j}) - F(t_1,\ldots, t_d)\right],$$

where $F$ is the joint distribution function of $X_0$. The factor-projected empirical process extends this classical paradigm by considering statistics or processes indexed by functionals $f$ belonging to a "projection" (or factor) space $\mathcal{G}$, often a subspace of Hölder, Lipschitz, or other smooth functions: $$G_n(f) = \frac{1}{\sqrt{n}} \sum_{i=1}^n \left[ f(X_i) - \mathbb{E} f(X_0) \right],$$ with two key modifications:

1. The function class $\mathcal{G}$ is selected so that indicator functions can be well-approximated by elements in $\mathcal{G}$ (via a control or modulus function $\tau$).
2. Limit theorems and tightness are established for $\{ G_n(f): f\in\mathcal{G} \}$, rather than directly for the original empirical process indexed by indicator functions.

This structure provides a bridge between empirical process theory and modern factor analysis, regression with dimension reduction, and goodness-of-fit testing for models with latent structures.

2. Weak Convergence via Projection and Smoother Function Classes

The general strategy for establishing weak convergence of the projected empirical process rests on three core conditions (Durieu et al., 2011):

• A. CLT in $\mathcal{G}$: For all $f \in \mathcal{G}$ of small seminorm, $(1/\sqrt{n}) \sum_{i=1}^n f(X_i)$ obeys a central limit theorem,

$$\frac{1}{\sqrt{n}}\sum_{i=1}^n f(X_i) \xrightarrow{d} N(0, \sigma_f^2),$$

even when $X_i$ exhibits strong serial dependence, e.g., in dynamical systems or Markov chains.

• B. Approximation of Indicators: For any rectangle $[a, b]\subset \mathbb{R}^d$, the indicator $1_{(-\infty, b]}$ can be sandwiched between functions $Y(a, b)\in\mathcal{G}$ with controlled seminorm $$\|Y(a,b)\|_{\mathcal{G}} \leq \tau\left(\min_{i}(F_i(b_i)-F_i(a_i))\right)$$.
• C. Uniform Moment Bounds: High moments of $G_n(f)$ are uniformly bounded in $f \in \mathcal{G}$, even when the dependence is only polynomially decaying.

Combining these, one proves (via a smoothing and approximation argument) that $U_n(t)$ converges weakly in $D([-\infty,\infty]^d)$ to a centered Gaussian process, as soon as the corresponding results hold for the projected class $\mathcal{G}$.

An explicit realization is given when $\mathcal{G}$ comprises bounded Hölder functions, making the abstract argument practically applicable for a broad class of processes.

3. Applications in Dynamical Systems, Markov Chains, and Factor Models

The factor-projection method is especially potent in systems with latent or observable factor structure:

• Dynamical Systems/Markov Chains: If the forward operator (transfer, Markov, or Perron–Frobenius operator) has a spectral gap on a smooth function space (e.g., Hölder or Lipschitz), central limit theorems and moment bounds hold for all $f$ in this space. Indicator functions rarely belong to the space, but they can be uniformly approximated using elements of $\mathcal{G}$—allowing the factor-projected empirical process theory to cover a broad set of dependent dynamical scenarios (Durieu et al., 2011).
• Semiparametric Factor Models: In settings where $X_t = \Lambda F_t + \varepsilon_t$ and $\Lambda$ is decomposed as $G(X) + \Gamma$ with $G$ governed by observable covariates (e.g., via sieve expansions), empirical process methods can be combined with projections $$Q = \Phi(X) (\Phi(X)^\top \Phi(X))^{-1} \Phi(X)^\top$$, yielding projected principal component analysis and improved estimation rates (Fan et al., 2014).
• High-Dimensional Matrix Factor Models: In matrix factor models $X_t = R F_t C^\top + E_t$, projecting data onto estimated factor spaces (either row or column) before eigen-analysis filters out noise more efficiently, stabilizing convergence of estimated loadings and factors (Yu et al., 2020).
• Regression with Sufficient Dimension Reduction: Empirical processes built from residuals, projected via estimated reduction directions, underlie advanced adaptive and omnibus model checking procedures in regression (Tan et al., 2016). Similarly, for functional linear models, random projections of high-dimensional or functional covariates enable powerful and computationally tractable tests (Cuesta-Albertos et al., 2017).

4. Extensions: Nonstationarity, Dependence, and Maximal Inequalities

• Locally Stationary Processes: The norm used to quantify the function class must incorporate both variance and local dependence structure (via a functional dependence measure and time-varying weighting). For $f(z, u) = D_{f, n}(u)\bar{f}(z, u)$, maximal inequalities and functional CLTs remain valid provided the "factor" (here $D_{f, n}(u)$) is controlled and the bracketing entropy of $F$ is finite (Phandoidaen et al., 2020).
• Nonsmooth Functions and Functional Dependence: The martingale difference decomposition, combined with functional dependence measures, allows working with nonsmooth functions (e.g., indicators, kernel bumps) and yields CLTs and uniform convergence rates under conditions weaker than strong mixing (e.g., polynomial decay suffice) (Phandoidaen et al., 2021). The maximal deviation of the empirical process indexed by factor-projected function classes is bounded as

$$\mathbb{E} \max_{f \in F} |G_n(f)| \leq c [ D_n r(\sigma / D_n) \sigma + q^* \frac{M^2 H}{n(D_n^\infty)^2 C_\Delta} ]$$

where $D_n$ is the weighting from the projection factor.

• Sampling Along Factor Trajectories: When empirical processes are sampled along nontrivial subsequences or random fields projected by ergodic sums, quantitative controls on "local times" and self-intersections (e.g., $V_n$, $M_n$ as functions of the sampling sequence) govern both the law of large numbers and FCLT properties for the projected process (Cohen et al., 2023).

5. Statistical Inference: Goodness-of-Fit and Variance Reduction

• Testing for Model Structure: Projection-based empirical process methods underpin adaptive and omnidirectional goodness-of-fit tests, enabling valid inference for models where the effective dimension is unknown or multi-index alternatives exist (Tan et al., 2016, Cuesta-Albertos et al., 2017). Martingale transformation and supremum over projection directions facilitate distribution-free and robust inference.
• Optimal Use of Auxiliary Information: When auxiliary variables or moment constraints are known, incorporating them via information-geometric projections (calibration, raking, empirical likelihood) yields an "informed" empirical process with strictly reduced variance for any functional that is correlated with the auxiliary information (Arradi-Alaoui, 2021). Explicitly, if $g$ is an auxiliary function, the limiting variance is reduced from $\mathbf{Var}_P(f)$ to $$\mathbf{Var}_P(f) - \mathrm{cov}_P(g, f)^\top \Sigma^{-1} \mathrm{cov}_P(g, f) \leq \mathbf{Var}_P(f)$$.

6. Limitations, Open Problems, and Broader Implications

• Dependence on Approximating Class: The performance and scope of the factor-projected empirical process depend critically on the ability to approximate relevant indicator functions with smooth functions in the chosen class $\mathcal{G}$ and on verifying central limit theorems and moment bounds for these classes.
• Mixing Conditions: Relaxing from exponential to polynomial mixing broadens applicability, but typically requires more involved moment and approximation controls.
• Extensions to Multimode and Multi-Factor Models: Projected estimation in matrix and tensor-valued data continues to advance, offering improved rates and interpretability, especially in domains such as finance and macroeconomics (Yu et al., 2020).
• Integration with Flexible Copula Structures: Incorporating flexible dependence modeling (e.g., vine copulas for factors (Han et al., 15 Aug 2025)) relies on uniform convergence of projected kernel density estimators for the margins, and a rigorous asymptotic theory for the factor-projected empirical process under joint serial and cross-sectional dependence is essential for consistent likelihood-based estimation.

7. Summary Table: Core Aspects of Factor-Projected Empirical Process

Aspect	Description	Example References
Projection/Factor Space	Function class $\mathcal{G}$ (e.g., Hölder, Lipschitz), sieve span, or row/column space in a matrix model	(Durieu et al., 2011, Fan et al., 2014)
Weak Convergence Goals	CLTs and Donsker property for $\{G_n(f): f \in \mathcal{G}\}$, uniform over $f$	(Durieu et al., 2011, Phandoidaen et al., 2020)
Applications	Dynamical systems, Markov chains, semiparametric loading, model checking, variance reduction, VaR forecasting	(Yu et al., 2020, Arradi-Alaoui, 2021, Han et al., 15 Aug 2025)
Statistical Benefits	Improved rates, robustness under dependence, variance reduction, higher power in model checking	(Fan et al., 2014, Tan et al., 2016)
Limiting Factors	Approximation error, moment control, rate of dependence decay, accuracy of factor/rotation estimation	(Durieu et al., 2011, Phandoidaen et al., 2021)

The factor-projected empirical process framework unifies a variety of advances in the modern theory of stochastic processes under complex dependence, high dimensions, and partial observability. It enables transfer of classical empirical process theory into settings with nontrivial (often latent) factor or projection structure, yielding both rigorous limit theorems and enhanced statistical methodology across time series, econometrics, and high-dimensional inference.