# Markov Chains - 2 clarification questions

I'm just getting started with Markov chains and have a few simple questions:

1. Is it possible to define a period for a reducible Markov chain? If so, how?
2. Can we define balance equations and a stationary distribution for reducible Markov chains?

Thanks.

Proposition. The state space $S$ of any Markov chain may be partitioned as $S = T \cup \bigcup_i C_i$, where the $C_i$ are closed and irreducible, and for every $x \in T$ there exists $y \in S$ such that $x \to y$ but $y \not\to x$.
So each $C_i$ can be considered an irreducible Markov chain in its own right. Each $C_i$ has a well-defined period $p_i$, but of course these periods may be different for different $i$. Each $C_i$ has at most one stationary distribution; if $C_i$ has stationary distribution $\pi_i$, it extends to a stationary distribution for $S$ by setting $\pi_i(x)=0$ for $x \notin C_i$. Thus the chain has (potentially) one stationary distribution for each $C_i$, and one can get more of them by taking convex combinations $\pi = \sum_i a_i \pi_i$ where $a_i \ge 0$, $\sum_i a_i = 1$.
The states of $T$ are all transient, so they don't really enter into the limiting behavior of the chain. For instance, any stationary distribution $\pi$ for $S$ must have $\pi(x) = 0$ for $x \in T$.