S Irreducible classes. Let E be countable, … a stochastic matrix on E, and Erec the set of all recurrent states. Let us say a state y is accessible from x, written as x ! y, if there exists some k  0 such that …k.x; y/ > 0. Show the following:

(a) The relation ‘!’ is an equivalence relation on Erec. The corresponding equivalence classes are called irreducible classes.

(b) If x is positive recurrent and x ! y, then y is also positive recurrent, and Ex

 Px nD1 1¹XnDyº



D Ex.x/=Ey.y/ :

In particular, all states within an irreducible class are of the same recurrence type.