Sharp FCLT in Wasserstein-1

• The paper establishes that the scaled integrals of stationary Gaussian processes converge to Brownian motion with an optimal rate of O(k⁻¹/²√ln k) in the Wasserstein-1 metric.
• It employs coupling techniques, duality with 1-Lipschitz test functions, and Gaussian supremum bounds to derive matching upper and lower bounds.
• The result highlights improved convergence properties in Wasserstein-1 compared to the Lévy–Prokhorov and bounded-Lipschitz metrics, emphasizing metric-dependent effects in functional convergence.

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 (Lototsky, 2022).

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 $W_1$0 and $W_1$1 is the set of all couplings. In this context, the bounded-Lipschitz metric $W_1$2 and the Lévy–Prokhorov metric $W_1$3 satisfy

$W_1$4

Thus, convergence in $W_1$5 implies weak convergence, and the rate in $W_1$6 controls the rate in $W_1$7.

2. Model Classes and Process Interpolations

The FCLT analysis considers two stationary Gaussian input models:

• Continuous-time Ornstein–Uhlenbeck: $W_1$8, stationary law $W_1$9, covariance $O(k^{-1/2}\sqrt{\ln k})$0.
• Discrete-time AR(1): $O(k^{-1/2}\sqrt{\ln k})$1, $O(k^{-1/2}\sqrt{\ln k})$2, with $O(k^{-1/2}\sqrt{\ln k})$3 ensuring $O(k^{-1/2}\sqrt{\ln k})$4, $O(k^{-1/2}\sqrt{\ln k})$5.

For $O(k^{-1/2}\sqrt{\ln k})$6, the scaled partial integrals or Donsker–interpolated processes are defined by

• $O(k^{-1/2}\sqrt{\ln k})$7
• $O(k^{-1/2}\sqrt{\ln k})$8

Let $O(k^{-1/2}\sqrt{\ln k})$9 be the law of $(E, d)$0 and $(E, d)$1 that of standard Brownian motion $(E, d)$2 on $(E, d)$3.

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 $(E, d)$4:

• Continuous-time case: There exist constants $(E, d)$5 so that for all $(E, d)$6,

$(E, d)$7

• Discrete-time AR(1) case: For $(E, d)$8, there exist $(E, d)$9 with analogous bounds.

Thus, the $E = C[0,1]$0 convergence rate is $E = C[0,1]$1. By contrast, in the Lévy–Prokhorov metric, the fastest achievable rate is $E = C[0,1]$2, so convergence in $E = C[0,1]$3 is improved by a factor of $E = C[0,1]$4.

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 $E = C[0,1]$5 as $E = C[0,1]$6.
• Duality with Lipschitz test functions: For any $E = C[0,1]$7-Lipschitz $E = C[0,1]$8,

$E = C[0,1]$9

• Gaussian-supremum theory: Using the Fernique–Sudakov and Borell–TIS inequalities, $\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$0.

For the lower bound, a specifically chosen $\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$1-Lipschitz test function $\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$2 exploiting the Gaussian maximal inequality yields a matching lower order, confirming the optimality of the bound.

5. Relations with Other Metrics

$\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$3 convergence is strictly stronger than convergence in the bounded-Lipschitz ($\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$4) and Lévy–Prokhorov ($\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$5) metrics, yet the sharpness of the convergence rate may fail in $\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$6. Table 1 summarizes the best known rates:

Metric	Convergence Rate	Sharpness Proven?
Wasserstein-1 ($\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$7)	$\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$8	Yes
Lévy–Prokhorov ($\|f\|_{\infty} = \sup_{0 \leq t \leq 1}|f(t)|$9)	$\mu$0	Yes
$\mu$1-Wasserstein	$\mu$2	Yes

A plausible implication is that $\mu$3 is sensitive to the sup-norm geometry of $\mu$4, resulting in log-improved convergence rates over $\mu$5 and $\mu$6, but the $\mu$7-Wasserstein rate does not require the logarithmic correction.

6. Extensions and Corollaries

The rate $\mu$8 for $\mu$9 convergence extends:

• To path spaces $\nu$0 under $\nu$1, up to a constant factor depending on $\nu$2.
• To additive-noise ODEs of the form $\nu$3 (with $\nu$4 Lipschitz), yielding $\nu$5 where $\nu$6 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 $\nu$7 factor is intrinsic to the $\nu$8 with sup-norm case, reflecting the slow divergence of Gaussian maxima over expanding time intervals. In weaker topologies (such as $\nu$9 or fractional-Sobolev norms), this logarithmic term can be eliminated, resulting in pure $E$0 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 (Lototsky, 2022).