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On weak convergence of Gaussian conditional distributions

Sarah Lumpp, Mathias Drton

TL;DR

This work shows that weak convergence of Gaussian joint laws does not, in general, imply convergence of conditional distributions when the limit covariance is singular. It introduces a precise set of conditions under which conditional covariances converge despite full-rank approximations, relying on a rank-$r$ decomposition of $\\Sigma_{S,S}$ and a nonzero asymptotic determinant coefficient $f_{\\mathrm{asy}}$; these conditions ensure $\\Sigma^{(m)}_{R|S} \\to \\Sigma_{R|S}$. The authors develop a generalized determinant framework and use Cramér’s rule to control the asymptotics, with extensions to non-symmetric cases. They illustrate the approach through Toeplitz-structured examples and an application to conditional independence in certain graphical models, highlighting how the method enables reasoning about conditional relationships where classical continuity fails.

Abstract

Weak convergence of joint distributions generally does not imply convergence of conditional distributions. In particular, conditional distributions need not converge when joint Gaussian distributions converge to a singular Gaussian limit. Algebraically, this is due to the fact that at singular covariance matrices, Schur complements are not continuous functions of the matrix entries. Our results lay out special conditions under which convergence of Gaussian conditional distributions nevertheless occurs, and we exemplify how this allows one to reason about conditional independence in a new class of graphical models.

On weak convergence of Gaussian conditional distributions

TL;DR

This work shows that weak convergence of Gaussian joint laws does not, in general, imply convergence of conditional distributions when the limit covariance is singular. It introduces a precise set of conditions under which conditional covariances converge despite full-rank approximations, relying on a rank- decomposition of and a nonzero asymptotic determinant coefficient ; these conditions ensure . The authors develop a generalized determinant framework and use Cramér’s rule to control the asymptotics, with extensions to non-symmetric cases. They illustrate the approach through Toeplitz-structured examples and an application to conditional independence in certain graphical models, highlighting how the method enables reasoning about conditional relationships where classical continuity fails.

Abstract

Weak convergence of joint distributions generally does not imply convergence of conditional distributions. In particular, conditional distributions need not converge when joint Gaussian distributions converge to a singular Gaussian limit. Algebraically, this is due to the fact that at singular covariance matrices, Schur complements are not continuous functions of the matrix entries. Our results lay out special conditions under which convergence of Gaussian conditional distributions nevertheless occurs, and we exemplify how this allows one to reason about conditional independence in a new class of graphical models.
Paper Structure (7 sections, 6 theorems, 46 equations)

This paper contains 7 sections, 6 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Motivating application
  3. Proof of the theorem
  4. An application involving Toeplitz matrices
  5. Proofs and additional calculations
  6. Proofs and calculations for Section 3
  7. Proofs for Section 4

Key Result

Theorem 1.2

Let $\Sigma^{(m)}$, $m\in\mathbb{N}$, be a sequence of invertible symmetric $p \times p$ matrices with singular limit $\Sigma$ and admitting the expansion Let $S\subset[p]$ with $k =|S|< p$, and let $R=[p]\setminus S$. If the matrices $\Sigma$ and $\Sigma^{(1,\infty)}$ satisfy that then the conditional covariance matrices $\Sigma^{(m)}_{R\mid S}$ converge to $\Sigma_{R\mid S}$.

Theorems & Definitions (19)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Example 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof : Proof of \ref{['thm:main_convergence']}
  • Lemma 4.1
  • ...and 9 more