Total variation convergence preserves conditional independence

Abstract

This note establishes that if a sequence $P_n,n=1,\dots$ of probability measures converges in total variation to the limiting probability measure $P$, and $\sigma$-algebras $A$ and $B$ are conditionally independent given $H$ 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.