Total Variation Convergence Preserves Conditional Independence

TL;DR

The paper addresses whether conditional independence of two σ-algebras given a third is preserved when the underlying probability measures converge. It proves a main result: convergence in total variation $P_n \to P$ preserves conditional independence, so $\mathbb{A} \perp\!\!\!\perp \mathbb{B} | \mathbb{H}$ under $P$ whenever it holds under all $P_n$. The proof analyzes the conditional probabilities $Z_n^A = P_n(A|\mathbb{H})$ in $L^2(P)$, extracts a weakly convergent subsequence, and identifies the limit as a version of $P(A|\mathbb{B} \vee \mathbb{H})$, ensuring the desired independence. A corollary shows that pointwise convergence of densities $f_n \to f$ (via Scheffé's theorem) also yields the same conditional independence in the limit. This result stabilizes conditional independence under strong forms of convergence and clarifies how density convergence interacts with independence properties.

Abstract

This note establishes that if a sequence $P_n, n=1,\ldots$ of probability measures converges in total variation to the limiting probability measure $P$, and $σ$-algebras $\mathbb{A}$ and $\mathbb{B}$ are conditionally independent given $\mathbb{H}$ with respect to $P_n$ for all $n$, then they are also conditionally independent with respect to the limiting measure $P$. As a corollary, this also extends to pointwise convergence of densities to a density.

Theorems & Definitions (4)

• Theorem 1
• proof
• Corollary 1
• proof