(Constructing Examples). Construct an example of each of the following phenomena:

(a) a Markov chain with only absorbing states;

(b) a Markov chain that is irreducible but not regular;

(c) a Markov chain that is irreducible but not aperiodic;

(d) a Markov chain on an infinite state space that is irreducible and aperiodic, but not regular;

(e) a Markov chain in which there is at least one null recurrent state;

(f) a Markov chain on an infinite state space such that every state is transient;

(g) a Markov chain such that each first-passage time Tij has all moments finite;

(h) a Markov chain without a proper stationary distribution;

(i) independent irreducible chains fXng; fYng, such that Zn D .Xn; Yn/ is not irreducible;

(j) Markov chains fXng; fYng such that Zn D .Xn; Yn/ is not a Markov chain.