Is the sum of the absolute values of the $X$ and $Y$ integer coordinates of a random walk in $R^2$ a Markov chain?
Ex. if the coordinates are ($X_n$, $Y_n$), and the probability of moving to any neighboring integer point is 1/4 (up, down, front, back). Is $Z_n$ = |$X_n$| + |$Y_n$| a Markov chain?