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

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 $L^2(σ)$ 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 $ρ$-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 $L^2(σ)$ 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.

Figures (2)

• Figure 1: Scenarios S1--S3 with asymmetric triangles. Residual location correlation $\mathrm{Corr}_{\mathrm{loc}}^{\mathrm{res}}$ separates clearly from the Steiner-point correlation, particularly in Scenario S3 where Steiner fails to detect strong size dependence.
• Figure 2: Scenario S4 (sensitivity analysis). The Steiner-point correlation remains close to zero for all $\alpha$, while the residual location correlation varies continuously, revealing directional location dependence invisible to Steiner-based summaries.

Theorems & Definitions (70)

• definition 3.1: Size/location/total covariance, variance, correlation
• Proposition 3.1: Structural orthogonality and additivity
• proof
• remark 3.1
• definition 3.2: Steiner--centered (residual) location component
• remark 3.2: Interpretation and role in dependence analysis
• remark 3.3: Steiner point as a projection of the odd component
• Proposition 3.2: Linearity, symmetry, scaling, translation invariance
• proof
• Proposition 3.3: Perfect sign flip for components