Irreducible MCs Why is it that theorems for (discrete) Markov chains always require that the MC concerned is irreducible? Can problems with reducible MCs can be simplified to considering the irreducible components? Thanks loads.
 A: You're right in the sense that in some cases the behavior of reducible MC can be studied through the study of its reducible components:  see e.g. discussion in Meyn, Tweedie 'Markov Chains and Stochastic Stability' in Section 4.1.3
The main idea is to introduce an equivalence relation $\sim$ on the state space $X$ as
$$
x\sim y\text{ iff there are m,n}\geq0 \text{ s.t. }P^m(x,y)>0\text{ and } P^n(y,x)>0
$$
where $P$ is a stochastic matrix and $P^0$ is an identity matrix. Such an equivalence relation makes a partition of the state space into equivalence classes $[x] = \{y\in X:y\sim x\}$. There is a countable set of such classes $C_1,C_2,\dots$ and some of them may be absorbing while others not. 
Being absorbing for a set $C_i\subseteq X$ means that if $x\in C_i$ then $P(x,y)>0$ implies $y\in C_i$. That means that there are no incoming transitions from other classes. Each of classes $C_i$ is irreducible and absorbing classes are sort of 'independent' from the other classes in the sense that there no transitions between elements of such classes. Hence they can be studied separately by applying the theory of irreducible Markov Chains: each absorbing irreducible class $C_i$ there is a correspondent irreducible Markov Chain which is obtained by considering only rows/columns of matrix $P$ related to $C_i$. 
Finally, the rest of classes of states are irreducible but not absorbing, i.e. they may have transactions to absorbing classes. On the other hand, for the study of infinite horizon properties of MC such as stability they usually are not important. E.g. one of the properties for such classes is that any finite subset will be left a.s. by a process in a finite time. For the finite-state Markov Chain that means that all non-absorbing equivalence classes $C_i$ will be left by a process in a finite time, so the process will escape to one of the absorbing equivalence classes. 
