= Solution
Every upward transition has positive probability. From any positive state the transition directly to zero has positive probability, since $1-\lambda/(i+1)>0$. Thus any state can reach zero and then reach any other state by successive upward steps, proving irreducibility.
If $\lambda<1$, the holding probability at zero is positive, so its period and hence the common period of the <irreducible> chain is one. At $\lambda=1$ that holding probability vanishes, but zero has positive-probability return paths $0\to1\to0$ of length two and $0\to1\to2\to0$ of length three. Their lengths have greatest common divisor one. Thus $\boxed{\text{the chain is irreducible and aperiodic for all }0<\lambda\le1}$.
Back to article page
