A characterization of mutual absolute continuity of probability measures on a filtered space

TL;DR

This paper introduces a new criterion for mutual absolute continuity of probability measures on filtered spaces using a martingale limit, linking it to Radon-Nikodym derivatives and tail behavior.

Contribution

It provides a novel characterization of mutual absolute continuity via a martingale limit and establishes convergence of Radon-Nikodym derivatives in $L^2$.

Findings

Mutual absolute continuity iff the martingale limit M equals 1 almost surely.

Radon-Nikodym derivatives converge in $L^2$ under the new characterization.

Application to families of random variables and stochastic processes.

Abstract

We give a new characterization for mutual absolute continuity of probability measures on a filtered space. For this, we introduce a martingale limit $M$ that measures the similarity between the tails of the probability measures restricted to the filtration. The measures are mutually absolutely continuous if and only if $M = 1$ holds almost surely for both measures. In this case, the square roots of the Radon-Nikodym derivatives on the filtration converge in $L^2$. Finally, we apply the result to families of random variables and stochastic processes.