---
title: Sharp FCLT in Wasserstein-1
url: https://www.emergentmind.com/topics/sharp-fclt-in-wasserstein-1
type: topic
---

# Sharp FCLT in Wasserstein-1

A sharp functional central limit theorem (FCLT) in the Wasserstein-1 metric quantifies the rate at which scaled integrals and Donsker-type interpolations of stationary Gaussian processes converge in law to Brownian motion, under the topology induced by the 1-Wasserstein distance on the space $C[0,1]$ of continuous functions equipped with the uniform norm. Unlike the classical finitely-dimensional CLT, the FCLT for such processes is nontrivial, and precise convergence rates reveal subtle metric-dependent phenomena that distinguish Wasserstein-1 from the Lévy–Prokhorov and bounded-Lipschitz metrics. Notably, the convergence rate in the 1-Wasserstein metric (“$W_1$”) is shown to be $O(k^{-1/2}\sqrt{\ln k})$, a rate that is slightly faster than the best known results for the Lévy–Prokhorov distance, establishing the exact sharp order for functional convergence in this metric [2209.08249].

## 1. Wasserstein-1 Metric on Path Space

Given a complete separable metric space $(E, d)$, specifically $E = C[0,1]$ with the uniform norm $\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$, the 1-Wasserstein distance between Borel probability measures $\mu$ and $\nu$ on $E$ is defined by both the dual (Kantorovich–Rubinstein) and primal (optimal coupling) representations:
\[
W_1(\mu, \nu) = \sup\left\{ |\int \phi\,d\mu - \int \phi\,d\nu| : \phi: E \to \mathbb{R},\ \text{Lip}(\phi) \leq 1 \right\} = \inf\left\{ \int d(x,y)\, \pi(dx,dy) : \pi \in \Pi(\mu, \nu) \right\},
\]
where $\text{Lip}(\phi) = \sup_{x \neq y}|\phi(x) - \phi(y)|/d(x,y)$ and $\Pi(\mu, \nu)$ is the set of all couplings. In this context, the bounded-Lipschitz metric $d_{BL}$ and the Lévy–Prokhorov metric $d_{LP}$ satisfy
\[
d_{BL}(\mu, \nu) \leq W_1(\mu, \nu), \quad d_{BL}(\mu, \nu) \leq 4d_{LP}(\mu, \nu) \leq 4W_1(\mu, \nu).
\]
Thus, convergence in $W_1$ implies weak convergence, and the rate in $W_1$ controls the rate in $d_{LP}$.

## 2. Model Classes and Process Interpolations

The FCLT analysis considers two stationary Gaussian input models:
- **Continuous-time Ornstein–Uhlenbeck:** $dX(t) = -2X(t)\,dt + 2\,dW(t)$, stationary law $N(0,1)$, covariance $\text{Cov}(X(s), X(t)) = e^{-2|t-s|}$.
- **Discrete-time AR(1):** $X_n = \alpha X_{n-1} + \epsilon_n$, $|\alpha| < 1$, with $\epsilon_n \sim \text{IID}\ N(0, (1-\alpha)/(1+\alpha))$ ensuring $\text{Var}\,X_n = 1$, $\sum_{k\in\mathbb{Z}}\text{Cov}(X_k, X_0) = 1$.

For $k \in \mathbb{N}$, the scaled partial integrals or Donsker–interpolated processes are defined by
- $W_k^{\text{cont}}(t) = k^{-1/2}\int_0^{kt} X(s) ds$
- $W_k^{\text{disc}}(t) = k^{-1/2} \sum_{n=1}^{\lfloor k t \rfloor} X_n + (k t - \lfloor k t \rfloor) X_{\lfloor k t \rfloor + 1}$

Let $\mu_k$ be the law of $W_k$ and $\mu_0$ that of standard Brownian motion $W$ on $C[0,1]$.

## 3. Sharp Rates in the Functional Central Limit Theorem

The primary result is an exact match of upper and lower bounds on the rate for $W_1(\mu_k, \mu_0)$:
- **Continuous-time case:** There exist constants $c_1, C_1 > 0$ so that for all $k \geq 1$,
  \[
  c_1\,k^{-1/2}\sqrt{\ln(1+k)} \leq W_1(\mu_k, \mu_0) \leq C_1\,k^{-1/2}\sqrt{\ln(1+k)}.
  \]
- **Discrete-time AR(1) case:** For $|\alpha| < 1$, there exist $c_2(\alpha), C_2(\alpha) > 0$ with analogous bounds.

Thus, the $W_1$ convergence rate is $O(k^{-1/2}\sqrt{\ln k})$. By contrast, in the Lévy–Prokhorov metric, the fastest achievable rate is $O((\ln k)/k^{1/2})$, so convergence in $W_1$ is improved by a factor of $1/\sqrt{\ln k}$.

## 4. Proof Outline and Key Technical Lemmas

The upper bound follows by a combination of coupling, duality arguments, and bounds on the maximum of Gaussian processes:
- **Gordin-type decomposition:** The scaled processes can be coupled to Brownian motion $W$ as $W_k(t) = k^{-1/2} W(kt) + k^{-1/2}(X(kt) - X(0))$.
- **Duality with Lipschitz test functions:** For any $1$-Lipschitz $\phi: C[0,1] \to \mathbb{R}$,
  \[
  |E[\phi(W_k)] - E[\phi(W)]| \leq E[\|X(k\,\cdot) - X(0)\|_\infty]\,k^{-1/2}
  \]
- **Gaussian-supremum theory:** Using the Fernique–Sudakov and Borell–TIS inequalities, $E[\sup_{0 \leq t \leq 1}|X(kt) - X(0)|]=O(\sqrt{\ln k})$.

For the lower bound, a specifically chosen $1$-Lipschitz test function $\phi_k$ exploiting the Gaussian maximal inequality yields a matching lower order, confirming the optimality of the bound.

## 5. Relations with Other Metrics

$W_1$ convergence is strictly stronger than convergence in the bounded-Lipschitz ($d_{BL}$) and Lévy–Prokhorov ($d_{LP}$) metrics, yet the sharpness of the convergence rate may fail in $d_{BL}$. Table 1 summarizes the best known rates:

| Metric                | Convergence Rate       | Sharpness Proven?     |
|-----------------------|-----------------------|-----------------------|
| Wasserstein-1 ($W_1$) | $O((\ln k)^{1/2}/k^{1/2})$ | Yes                  |
| Lévy–Prokhorov ($d_{LP}$) | $O((\ln k)/k^{1/2})$    | Yes                  |
| $L^1$-Wasserstein     | $O(k^{-1/2})$            | Yes                  |

A plausible implication is that $W_1$ is sensitive to the sup-norm geometry of $C[0,1]$, resulting in log-improved convergence rates over $d_{LP}$ and $d_{BL}$, but the $L^1$-Wasserstein rate does not require the logarithmic correction.

## 6. Extensions and Corollaries

The rate $O(k^{-1/2}\sqrt{\ln k})$ for $W_1$ convergence extends:
- To path spaces $C[0,T]$ under $\|\cdot\|_\infty$, up to a constant factor depending on $T$.
- To additive-noise ODEs of the form $dY_k = F(Y_k)\,dt + dW_k$ (with $F$ Lipschitz), yielding $W_1(\mathcal{L}(Y_k), \mathcal{L}(Y)) = O(k^{-1/2}\sqrt{\ln k})$ where $Y$ solves the analogous SDE driven by Brownian motion.
- Under martingale-difference or mixing hypotheses, Gordin’s decomposition with Gaussian-supremum techniques can yield analogous rates for certain non-Gaussian inputs.

## 7. Optimality and Limit Cases

The $\sqrt{\ln k}$ factor is intrinsic to the $C[0,1]$ with sup-norm case, reflecting the slow divergence of Gaussian maxima over expanding time intervals. In weaker topologies (such as $L^2[0,1]$ or fractional-Sobolev norms), this logarithmic term can be eliminated, resulting in pure $O(k^{-1/2})$ rates. A plausible implication is that the metric governing convergence has a decisive effect on the attainable rates in infinite-dimensional settings. Extending sharp FCLT rates to non-stationary or heavy-tailed Gaussian processes, or to Gaussian processes indexed by higher-dimensional parameter spaces, remains an open frontier [2209.08249].

Source: https://www.emergentmind.com/topics/sharp-fclt-in-wasserstein-1