Contiguity of probability measures ($Q_n◃P_n$)

A sequence $Q_n$ is contiguous with respect to $P_n$ if $P_n(A_n)\to 0$ implies $Q_n(A_n)\to 0$ for every measurable sequence of events. It transfers negligible-event assertions, including consistency, between changing sampling laws. It need not imply small total variation distance.

Let $L_n$ be the probability density function of the absolutely continuous part of $Q_n$ relative to $P_n$. If $L_n$ converges in distribution under $P_n$ to $L>0$ almost surely with $\mathbb{E}L=1$, then $P_n,Q_n$ are mutually contiguous. Indeed, $\mathbb{E}{L}_n\le 1$ and convergence to a mean-one limit imply uniform integrability and vanishing singular mass, giving forward contiguity. For reverse contiguity, $P_n(A_n)\le P_n(L_n\le c)+Q_n(A_n)/c$, and one first takes $n\to \infty$ and then $c\downarrow 0$.

Two sequences of probability laws are mutually contiguous when contiguity of probability measures holds in both directions. A positive mean-one limit of their likelihood ratio is a standard sufficient criterion.