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The pair $Z_n=(X_n,Y_n)$ is an irreducible positive recurrent product Markov chain with stationary distributionThe stationary cycle occupation formula says that, during one return cycle to a state $z$, the expected number of visits to a set $A$ is $\Pi (A)/\Pi (z)$. TakeThenThe initial and terminal states both have $Y=0$, so either convention for including the endpoints gives the same count. Thus

Take $d$ independent copies of the simple random walk on the integer line, each stepping left or right with equal probability at every integer time. Their product Markov chain moves all coordinates at once; each of its $2^d$ diagonal increments has probability $2^{-d}$. At odd times a return to the origin is impossible. At time $2n$, independence gives the displayed return probability, whose asymptotic equivalence follows from Stirling's approximation. The recurrence criterion by return probabilities and the p-series therefore imply recurrence for $d=1,2$ and a transient Markov chain for $d\ge 3$.

For $d>1$ the chain from the origin visits only the sublattice whose coordinates all have the same parity. This is one communicating class; restricting to it gives an irreducible Markov chain. The period of a state in a Markov chain is two here. The conclusion applies to that class, without incorrectly claiming Markov chain irreducibility on all of ${\mathbb{Z}}^d$. This walk moves diagonally; the usual axis-step simple random walk has different transition probabilities, even though the same dimensional recurrence threshold holds.