On Linear Estimators for some Stable Vectors

TL;DR

The paper addresses estimating a symmetric $S\alpha S$ variable $X$ from an observation $Y$ under two dependency models: a linear-mix of independent $S\alpha S$ components and an elliptically contoured sub-Gaussian $S\alpha S$ vector. It proves that the conditional mean $\mathsf{E}[X|Y=y]$ is linear in $Y$ in both models and derives dispersion-optimal linear estimators, showing that these estimators differ from the Gaussian case in the linear-mix model but coincide with the Gaussian result in the sub-Gaussian model. In the sub-Gaussian setting, the conditional mean, dispersion-optimal linear estimator, and MAP estimator all align with a single coefficient $\beta=\rho\frac{\sigma_{G_1}}{\sigma_{G_2}}$, highlighting an elegant heavy-tailed extension of Gaussian linear estimation. Overall, the work clarifies when linear estimators remain Bayes-optimal under heavy-tailed dependence and identifies elliptical sub-Gaussian $S\alpha S$ as the natural heavy-tailed analogue to Gaussian linear estimation.

Abstract

We consider the estimation problem for jointly stable random variables. Under two specific dependency models: a linear transformation of two independent stable variables and a sub-Gaussian symmetric $α$-stable (S$α$S) vector, we show that the conditional mean estimator is linear in both cases. Moreover, we find dispersion optimal linear estimators. Interestingly, for the sub-Gaussian (S$α$S) vector, both estimators are identical generalizing the well-known Gaussian result of the conditional mean being the best linear minimum-mean square estimator.

Theorems & Definitions (7)

• Theorem 1: Linear Mixture Conditional Expectation
• Theorem 2: Dispersion-Optimal Linear Estimator
• Lemma 1: The Conditional Representation
• Lemma 2: Integrability of the residual term
• Corollary 1: Conditional Mean is Linear in $Y$
• Theorem 3: Optimal Linear Coefficient under a Dispersion Constraint (sub-Gaussian version)
• Theorem 4: MAP optimality