Uniform growing-state and learned-model extensions

Establish rates uniform over growing state counts K=K(q) and extend the vanishing-predictive-KL phenomenon to learned sequence models.

Background

The main theorems fix the number of latent states K before taking the rare-switching limit q→0, and their constants may deteriorate as K grows or emission means become nearly coincident. The construction also concerns explicit mathematical filters rather than models obtained through training. Consequently, the paper leaves unresolved both a joint or uniform asymptotic theory when the state count grows with q and extensions of the phenomenon to learned recurrent or state-space sequence models.