Convergence of the adapted empirical measure for mixing observations

TL;DR

The paper develops a rigorous theory for estimating the law of a time-dependent stochastic process in the adapted Wasserstein space when observations are mixing rather than iid. It introduces the adapted empirical measure $\widehat{\mu}^N$ and derives moment bounds, sub-exponential concentration, and almost-sure consistency under a generalized $\eta$-mixing framework, along with new bounded-differences-type inequalities for uncountable spaces. The results unify compact and noncompact settings, employing discretization, smoothing, and compact-approximation to extend known iid rates to dependent data. Practical implications include robust, dynamically consistent inference for time-dependent optimization problems in finance and operations research where dependence is present. The numerical experiments corroborate the theoretical rates and illustrate the impact of mixing on convergence behavior.

Abstract

The adapted Wasserstein distance $\mathcal{AW}$ is a modification of the classical Wasserstein metric, that provides robust and dynamically consistent comparisons of laws of stochastic processes, and has proved particularly useful in the analysis of stochastic control problems, model uncertainty, and mathematical finance. In applications, the law of a stochastic process $μ$ is not directly observed, and has to be inferred from a finite number of samples. As the empirical measure is not $\mathcal{AW}$-consistent, Backhoff, Bartl, Beiglböck and Wiesel introduced the adapted empirical measure $\widehatμ^N$, a suitable modification, and proved its $\mathcal{AW}$-consistency when observations are i.i.d. In this paper we study $\mathcal{AW}$-convergence of the adapted empirical measure $\widehatμ^N$ to the population distribution $μ$, for observations satisfying a generalization of the $η$-mixing condition introduced by Kontorovich and Ramanan. We establish moment bounds and sub-exponential concentration inequalities for $\mathcal{AW}(μ,\widehatμ^N)$, and prove consistency of $\widehatμ^N$. In addition, we extend the Bounded Differences inequality of Kontorovich and Ramanan for $η$-mixing observations to uncountable spaces, a result that may be of independent interest. Numerical simulations illustrating our theory are also provided.

Figures (4)

• Figure 1: Observations $\widehat{\bold{X}}:=(X^n)_{n = 1}^N \in (\mathbb{R}^d)^T$.
• Figure 2: Sample paths of $\mu^N$ (left) and $\widehat{\mu}^N$ (right) in the case $d = 1$ and $T = 3$. Some trajectories are merged as their values belong to the same interval, hence the corresponding conditional distributions are non-degenerate.
• Figure 3: Rate of convergence of $\mathbb{E}\,\mathcal{AW}(\mu, \widehat{\mu}^N)$ for $D = 2, 5, 10$ compared to the theoretical upper bound $C \cdot \sqrt{1 + 2 \sum_{s = 1}^{N-1} \eta_{\widehat{\bold{X}}}(s)} \cdot N^{-1/3}$ from Theorem \ref{['thm:moment_estimate_compact']}. The expected value is estimated using empirical average over $M = 5000$ independent samples.
• Figure 4: Empirical distribution of $\mathcal{AW}(\mu, \widehat{\mu}^N)$ for $D = 2, 5, 10$ over $M = 5000$ samples for $N = 5000$.

Theorems & Definitions (90)

• Definition 3.1
• Definition 3.2: cf. bartl2021wasserstein
• Definition 3.3
• Example 3.4
• Example 3.5
• Definition 3.6
• Remark 3.7
• Proposition 3.8
• proof
• Remark 3.9