Intro

Definition (Weak Merging in expectation)

A filter process is stable in the sense of weak merging in expectation if for any $f\in C_{b}(\mathbb{X})$ and any prior $\nu$ with $\mu\ll\nu$ we have $$\lim_{ n \to \infty } \mathbb{E}^{\mu}\left[ \left| \int\limits f \, d\pi_{n}^{\mu}-\int\limits f \, d\pi_{n}^{\nu} \right| \right]=0$$

Definition (Weak merging almost surely)

A filter process is stable in the sense of weak merging $\mathbb{P}^{\mu}$ a.s. if there exists a set of measurement sequences $A\subset \mathbb{Y}^{\mathbb{Z}_{+}}$ with $\mathbb{P}^{\mu}$ probability $1$ such that for any sequence in $A$, for any $f\in C_{b}(\mathbb{X})$ and any prior $\nu$ with $\mu\ll\nu$, we have $$\lim_{ n \to \infty } \left| \int\limits f \, d\pi_{n}^{\mu}-\int\limits f \, d\pi_{n}^{\nu} \right| =0\quad \mathbb{P}^{\mu}\text{ a.s.}$$

Definition (Total Variation metric in expectation)

A filter process is stable in the sense of total variation in expectation if for any prior $\nu$ with $\mu\ll\nu$, we have $$\lim_{ n \to \infty } \mathbb{E}[\lVert \pi_{n}^{\mu}-\pi_{n}^{\nu} \rVert_{TV} ]=0$$

Definition (Total Variation metric almost surely)

A filter process is stable in the sense of total variation $\mathbb{P}^{\mu}$ a.s. if for any prior $\nu$ with $\mu\ll\nu$ we have $$\lim_{ n \to \infty } \lVert \pi_{n}^{\mu}-\pi_{n}^{\nu} \rVert _{TV}=0\quad\mathbb{P}^{\mu}\text{ a.s.}$$

Definition (Divergence)

A filter process is stable in relative entropy if for any prior $\nu$ with $\mu\ll\nu$ $$\lim_{ n \to \infty } \mathbb{E}^{\mu}[D(\pi_{n}^{\mu}\Vert \pi_{n}^{\nu} )]=0$$

Definition (Bounded-Lipschitz metric merging)

A system is stable in the sense of BL-merging $\mathbb{P}^{\mu}$ a.s. if we have $$\lim_{ n \to \infty } \lVert \pi_{n}^{\mu}-\pi_{n}^{\nu} \rVert =0\quad \mathbb{P}^{\mu}\text{ a.s.}$$