 Solution

After a jump from one to $k$, the first return to one occurs exactly $k$ steps later, with probability ${\alpha }_k$. Later return times are sums of such positive-support values. Thus their greatest common divisor isIn an irreducible chain this is the common period of every state. Hence the chain is aperiodic exactly when $d=1$. A self-loop at one is sufficient but not necessary; support at two and three also has greatest common divisor one.