Let $\displaystyle Z_1, Z_2, ...$ be a Bernoulli Process with success parameter p. Prove if

$\displaystyle Y_0, Y_1,...$, where $\displaystyle Y_{k+1} = max(0, Y_k + 2Z_{k+1} - 1)$ for $\displaystyle k = 0,1,...$ and $\displaystyle Y_0 = 0$

and

$\displaystyle X_0, X_1,..$ where $\displaystyle X_{k+1} = Z_{k+1} + Z_k$ for $\displaystyle k = 0,1,...,$ and $\displaystyle X_0 = 0$

are Markov chains.