Mixture models lend themselves nicely to HMM (hidden Markov model) treatment. Obviously, HMM can be nonmixture or not resolve mixtures when there is only one resolvable input or the superposition of self-similar inputs. What I am asking for is whether or not there are any not so trivial examples of nonmixture HMM data, with the impact relating to an attempt to define HMM as non-trivially different than mixture model analysis, e.g., see https://stats.stackexchange.com/a/249712/99274 $\leftarrow$Let me ask, for example, if the example I gave is exactly correct or not.
Be mindful please that a hidden Markov model can be considered a generalization of a mixture model where the hidden variables (or latent variables), which control the mixture component to be selected for each observation, are related through a Markov process rather than independent of each other..
And moreover that In the standard type of hidden Markov model considered here, the state space of the hidden variables is discrete, while the observations themselves can either be discrete (typically generated from a categorical distribution) or continuous (typically from a Gaussian distribution).