Let $\{X_t\}_{t \geq 0}$ be two-state Markov Chain with state space $S=\{0,1\}$, transition matrix $$ P= \begin{bmatrix} 1-p & p \\ q & 1-q \end{bmatrix} $$ and initial distribution $\pi _0 = (\pi _0 (0),\pi_0 (1))$. Introduce new stochastic process $\{Y_t\}_{t\geq 1}$ and $\{Z_t\}_{t\geq 1}$ by setting $Y_t=X_t+X_{t-1}$ and $Z_t=10X_t+X_{t-1}$.
Are either of $\{Y_t\}_{t\geq 1}$ or $\{Z_t\}_{t\geq 1}$ Markov Chains ?
My attempt:
$\{Y_t\}_{t\geq 1}$ is not a Markov Chain as $$\mathbb{P}(Y_3=0|Y_{2}=1,Y_1=1)\neq \mathbb{P}(Y_2=0|Y_2=1,Y_1=0) $$ Was not able to find a counterexample for $\{Z_t\}_{t\geq 1}$
Looking for :
1) An intuitive reasoning as to why $\{Y_t\}_{t\geq 1}$ is not a Markov Chain.
2) Whether or not $\{Z_t\}_{t\geq 1}$ is a Markov Chain and an intuitive reasoning for it.
3) Your thought process in determining whether or not a stochastic process is a Markov Chain.
Any insight is greatly appreciated, thank you.