Let $H_1$ and $H_2$ be two instances of a finite Hidden Markov Model (HMM) $H$. That is, $H_1$ and $H_2$ have identical state spaces $Q$ as well as identical transition $A$ and emission probabilities $B$. However, they have different initial distributions $\pi_1$ and $\pi_2$, respectively. Let $(O_0, O_1, O_2\ldots)$ be a sequence of observations applied to both $H_1$ and $H_2$.
Has anyone considered the question how long does it take both $H_1$ and $H_2$ to forget their initial distributions? That means their belief states $b_1: Q \rightarrow [0, 1]$ and $b_2: Q \rightarrow [0, 1]$ become arbitrarily close to each other?
By "arbitrarily close" here I mean "Given any $\epsilon > 0$, what is $N(\epsilon) > 0$ such that, for all observations $O_n, n > N(\epsilon):\ dist(b_1, b_2) < \epsilon$ for some distance function $dist$?"