 Solution

A finite irreducible aperiodic Markov chain converges to its unique stationary distribution from every initial state. Finiteness holds because nonnegative entries with fixed finite row sums are bounded. The given connecting-sequence property makes the chain irreducible: each feasible step in such a sequence has positive proposal probability and is accepted. The all-zero index table occurs with probability $1/L>0$ and gives a self-loop at every state, establishing aperiodicity. Reversibility supplied the stationary uniform law in (b). ConsequentlyThe conclusion also holds when the feasible state space is a singleton.