Consider a process $\{x_n,n=0,1,$dots$\},$ which takes on the values $0,1,$ or $2.$ Suppose

$P\{x_{n+1}=j|x_n=i,x_{n-1}=i_{n-1},$dots,$x_0=i+0\}=\{\begin{array}{ccc}{P}_{ij}^{\text{I}}& \text{w}\text{h}\text{e}\text{n}nis\text{e}\text{v}\text{e}\text{n}& \\ P_{ij}^{II}& \text{w}\text{h}\text{e}\text{n}nis\text{o}\text{d}\text{d}& \end{array}$

where ${\displaystyle \underset{j=0}{\overset{2}{}}}{P}_{ij}^{\text{I}}={\displaystyle \underset{j=0}{\overset{2}{}}}{P}_{ij}^{II}=1,i=0,1,2.$ Is $\{x_n,n=0,1,$dots$\}$ a time$-$homogeneous Markov

chain? If not, then show how, by enlarging the state space, we may transform it into a time$-$

homogeneous Markov chain.