Laws of Large Numbers for Information Resolution

TL;DR

The paper develops a framework for laws of large numbers governing the recovery of information resolution, encoded by the σ-field, from iid samples. It defines empirical resolution σ-fields and proves monotone convergence to the full σ-field in $\\mathbb{R}^d$ and in general metric spaces under Vitali-type regularity, with finite-sample $L^1$ bounds for Lipschitz test functions. The results enable two key applications: constructing randomized Skorokhod embeddings via refining partitions and analyzing the risk of randomized regression trees in random forests through partition-based information resolution. This work bridges probabilistic σ-field recovery with practical partition-based methods, providing both theoretical convergence guarantees and actionable bounds for learning with resolution-based estimators.

Abstract

Laws of large numbers establish asymptotic guarantees for recovering features of a probability distribution using independent samples. We introduce a framework for proving analogous results for recovery of the $σ$-field of a probability space, interpreted as information resolution--the granularity of measurable events given by comparison to our samples. Our main results show that, under iid sampling, the Borel $σ$-field in $\mathbb R^d$ and in more general metric spaces can be recovered in the strongest possible mode of convergence. We also derive finite-sample $L^1$ bounds for uniform convergence of $σ$-fields on $[0,1]^d$. We illustrate the use of our framework with two applications: constructing randomized solutions to the Skorokhod embedding problem, and analyzing the loss of variants of random forests for regression.

Figures (10)

• Figure 1: Comparing a new sample $Z_i$ to the previous samples $X_1,X_2,X_3$.
• Figure 2: Approximating a box $A_q$ from inside in the proof of Theorem \ref{['classical R^n G-C']}.
• Figure 3: An adversarially chosen function which maximizes the Hausdorff distance.
• Figure 4: The points in $\widetilde{\mathcal{F}_n}$ splitting the unit cube in every coordinate.
• Figure 5: Top: first step of Dubins' binary splitting, with barriers $x_1$ and $x_2$. Bottom: refinement using $y_1,\dots,y_4$.

Theorems & Definitions (37)

• Example 1.1
• Example 1.2
• Definition 2.1
• Proposition 2.1
• proof
• Theorem 2.1
• Lemma 2.1
• proof : Proof of Lemma \ref{['inf-achieved']}
• Corollary 2.1: $L^p(\mu)$ Closure of $\sigma$-fields
• proof : Proof of Theorem \ref{['classical R^n G-C']}