Hidden Markov Models with Finite State Space and Infinite Observation Space

Question

How could I rewrite the following definition of a hidden Markov model for a finite signal state space $E$ and an infinite observation state space $F$? Or if both were finite?

A stochastic process $(X_{k}, Y_{k})_{k\ge 0}$ on a product state space $(E\times F, \mathcal{E}\otimes \mathcal{F})$ if there exist transition kernels $P: E\times \mathcal{E}\to [0,1]$ and $\Phi: E\times \mathcal{F}\to [0,1]$ such that $$ \mathbf{E}(g(X_{k+1},Y_{k_1})|X_{1}=x_{1},\ldots,X_{n}=x_{n})= \displaystyle \int_{E\times F} g(x,y) \Phi(x,dy)P(X_{k},dx) $$ and an initial measure $\mu$ on $E$ such that $$\mathbf{E}(g(X_0,Y_0))=\displaystyle\int_{E\times F}g(x,y)\Phi(x,dy)\mu(dx)$$ for any bounded measurable function $g:E\times F\to\mathbb{R}$.

Finally, is there a clear connection between this definition and the "standard" definition of discrete hidden markov models, $$\mathbf{P}(Y_n\in A|X_1=x_1,\ldots,X_n=x_n)=\mathbf{P}(Y_n\in A|X_n=x_n)$$ for every Borel set $A$.

Answer 1

The definition remains the same regardless of whether the observation space is finite or not. You can find a clear definition in classical references on HMMs such as the book Inference In Hidden Markov Models