Knight on chess board aperiodic?

Question

So suppose I have a knight on the corner of a chess board. I have to work out the average return time and also the limit of the probability that the knight is back in the corner after n steps as n tends to infinity. I have done this using a model of vertex and edges and using the stationary distribution, but the theorem I have used for the second part says that the distribution will converge to the stationary distribution if the markov chain is aperiodic and irreducible. I dont think this chain is aperiodic so surely the theorem would not apply?

Answer 1

It's true that the knight's walk is periodic; specifically, it has period 2. As such, any statement you can make about its limiting distribution will have to be one about the walk when sampled after an even number of steps; that is, you can address the distribution of $Y_n := X_{2n}$, where $X_n$ is the position of the walk after $n$ steps. Note that $Y_n$ is aperiodic, and hence the theorem applies. For the odd-numbered steps of $X_n$, the calculation is trivially $0$. (I'll leave as an exercise the details of what changes in the distributions when you consider $X_{2n}$ versus $X_n$.)

You may also find the theorem referenced in this question to be helpful.