Klenke gives a construction for a discrete Markov chain (Section 17.2 "Discrete Markov Chains: Examples", pp. 353-354). I don't understand several points in this construction, as indicated below.
The construction
Let $\left(R_n\right)_{n\in\mathbb{N}_0}$ be an independent family of random variables, all defined on the common probability space $\left(\Omega,\mathcal{A},\mathrm{P}\right)$ and each with values in $E^E$ (where $E$ is countable) and with the property $\left(*\right)\space\space\mathrm{P}\left[R_n\left(x\right)=y\right]=p\left(x,y\right)\space\space\mathrm{for}\space\mathrm{all}\space x,y\in E$ where $p\in E\times E\rightarrow\left[0,1\right]$ is a stochastic matrix.
For example, choose $\left(R_{n,x}\right)_{x\in E,n\in\mathbb{N}}$ as an independent family of random variables with values in $E$ and distributions $\left(**\right)\space\space\mathrm{P}\left[R_{n,x}=y\right]=p\left(x,y\right)\space\mathrm{for}\space\mathrm{all}\space x,y\in E\space\mathrm{and}\space n\in\mathbb{N}_0$
Note, however, that in $\left(*\right)$ we have required neither independence of the random variables $\left(R_{n,x}\right)_{x\in E}$ nor that all $\left(R_{n,x}\right)_{n\in\mathbb{N}_0}$ have the same distribution.
For $x\in E$ define $X_0^x=x\space\space\mathrm{and}\space\space X_n^x=R_{n,X_{n-1}^x}\space\space\mathrm{for}\space n\in\mathbb{N}$
Finally, let $\mathrm{P}_x:=\mathcal{L}\left[X^x\right]$ be the distribution of $X^x$. Recall that this is a probability measure on the space of sequences $\left(E^{\mathbb{N}_0},\mathcal{B}\left(E\right)^{\otimes\mathbb{N}_0}\right)$.
Theorem With respect to the distribution $\left(\mathrm{P}_x\right)_{x\in E}$, the canonical process on $\left(E^{\mathbb{N}_0}, \mathcal{B}\left(E\right)^{\otimes\mathbb{N}_0}\right)$ is a Markov chain with transition matrix $p$. In particular, to any stochastic matrix $p$, there corresponds a unique discrete Markov chain (namely, the canonical process) with transition probabilities $p$.
My questions
- Previously in the book (at the beginning of section 17.1 "Markov Chains: Definitions and Construction", p. 345), a Markov process's range set $E$ is required by definition to be a Polish space. In the construction above all that is assumed on $E$ is that it is countable. Is countability a sufficient condition for "Polishness"?
- How does $\left(**\right)$ (which is $\mathcal{A}-\mathcal{B}\left(E\right)^{\otimes\left(E\times\mathbb{N}_0\right)}$ measurable) define a process $\left(*\right)$ (which is $\mathcal{A}-\left(\mathcal{B}\left(E\right)^{\otimes E}\right)^{\otimes\mathbb{N}_0}$ measurable)?
- Why is $X_n^x$ a $\mathcal{A}-\mathcal{B}\left(E\right)$ measurable random object (for each $x\in E$, $n\in\mathbb{N}_0$)?