Assume sequence $(X_1,X_2, X_3, \ldots)$ is Markov sequence of real random variables where $X_i \in \mathcal{X}$ for some alphabet $\mathcal{X}$ of finite size $k$. Define random variable $Y_i = (X_{i-1},X_i)$ for $i \geq 2$. Is $Y_2, Y_3, \ldots$ a Markov sequnece?

In general, can we say that if $(A_1,A_2, A_3, \ldots)$ and $(B_1,B_2, B_3, \ldots)$ are Markov sequences where $A_i$ and $B_i$ coming from some finite alphabets, then $(C_1,C_2, C_3, \ldots)$ is Markov where $C_i = (A_i,B_i)$?