Size-Location Correlation for Set-Valued Processes: Theory, Estimation, and Laws of Large Numbers under $\rho$-Mixing

TL;DR

This paper develops a new theoretical framework for analyzing dependence in set-valued data using support functions and an even--odd decomposition, leading to novel measures and laws of large numbers.

Contribution

It introduces a variational approach with size-location decomposition, new dependence measures, and establishes laws of large numbers for set-valued processes.

Findings

New size, location, and total covariance indices for random sets.

Dependence measures are geometrically interpretable and translation-invariant.

Laws of large numbers for Minkowski averages under weak stationarity.

Abstract

We propose a variational framework for analyzing dependence structures of convex compact random sets based on their support functions. The approach relies on the canonical even--odd decomposition on the unit sphere, which separates size-related and location-related components and induces an exact orthogonality in the sphere $L^2(\sigma)$ space. This decomposition yields an additive variance--covariance structure that is intrinsic to set-valued data and cannot be recovered from point-based or selection-based representations. Within this framework, we introduce size, location, and total covariance and correlation indices for random sets, together with compatible $\rho$-mixing coefficients for set-valued processes. The resulting dependence measures are geometrically interpretable, invariant under translations, and free of degeneracies that arise for centrally symmetric sets under classical approaches. Weak and strong laws of large numbers are established under weak stationarity, providing asymptotic stability of Minkowski averages in the $L^2(\sigma)$ support-function norm. The proposed quantities admit natural numerical realizations via directional Monte Carlo and spherical designs. Applications to interval-valued and convex-valued data, including regression with set-valued responses, illustrate how the even--odd decomposition disentangles directional location dependence from size effects beyond what can be captured by finite-dimensional summaries such as the Steiner point.