Consider a discrete-time Markov chain with state space S = {0, 1, 2, ...} and the following transition probabilities {p_ij}:
p_{i,i-1} = 1, for all i ≥ 1,
p_{0,i} = q_i, for all i ≥ 1,
with ∑_{i=1}^∑ q_i = 1 and q_i > 0 for all i ≥ 1. Determine
(1) whether the Markov chain is irreducible;
(2) which states are recurrent;
(3) whether the Markov chain is aperiodic;
(4) which states are positive recurrent;
(5) the mean return time for each state.