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

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.