# Do hidden Markov models contain Markov chains?

Is it correct to say that the Hidden State Sequence in a Hidden Markov Model is a Markov Chain?

Thanks

Yes, it's definitely a Markov chain because the observed variables define the transition probabilities from every latent variable configuration to every other. (That is, they yield an $n \times n$ matrix of probabilities where $n$ is the number of latent variable configurations.)
Yes, A hidden Markov model is a bivariate stochastic process $\{(X_t,Y_t)\}_{t\geq0}$, defined on a product space $(\mathsf{X} \times \mathsf{Y},\mathcal{X} \otimes \mathcal{Y})$, where $\{X_t\}_{t\geq0}$ is a Markov chain taking values in $\mathsf{X}$ and $\{Y_t\}_{t\geq0}$, defined on some state space $\mathsf{Y}$, is a sequence of observable random variables conditional on $\{X_t\}_{t\geq0}$.