Past exam of the probability course of the University of Oxford / 2014 / 3 / c

Represent the simple random walk by $X_n={\xi }_1+\cdots +{\xi }_n$, where the increments are independent and identically distributed and satisfy $\mathbb{P}({\xi }_k=1)=p$, $\mathbb{P}({\xi }_k=-1)=1-p$. They are integrable and have expectation $2p-1$, so the strong law of large numbers givesIf $p\ne 1/2$, this nonzero drift implies that $X_n$ eventually leaves every fixed finite interval and tends to the appropriate infinity. Hence it visits zero only finitely many times almost surely, so zero is a transient state. By translation of the starting state, every state is transient: the chain is transient for $p\ne 1/2$. This also covers the deterministic endpoint cases $p=0,1$, without assuming Markov chain irreducibility at those endpoints.

For the symmetric simple random walk on the integer line, odd return times are impossible. A return after $2m$ steps requires exactly $m$ positive increments, soby Stirling's approximation. The resulting sum diverges by the p-series criterion. The recurrence criterion by return probabilities therefore makes zero a recurrent state. The simple random walk is an irreducible Markov chain on $\mathbb{Z}$, so the symmetric chain is recurrent.

This is the synchronous product of simple symmetric random walks, rather than a walk that changes only one coordinate at a time. Independence of the three coordinate processes givesOdd-time return probabilities vanish. The p-series with exponent $3/2$ converges, so the sum of all return probabilities is finite. By the recurrence criterion by return probabilities, the origin is a transient state: the three-dimensional process is transient. From the origin its communicating class consists of integer triples whose coordinates have the same parity; translation gives the same classification for every other class.