Theorem 1.6

If  denotes the probability that the chain starting with a transient state,  eventually gets absorbed in an absorbing state, . Let us denote the absorption probability matrix by , , then

Proof of Theorem 1.6

Now we have since transition between absorbing states are impossible. Moreover . Since  is absorbing so once the chain reaches an absorbing state  after steps , thus
This is true as  can be any state in the Markov chain.

Note: Now we utilize the concept of ascending order of a sequence and its results, i.e.,  holds true for the case when . In case of descending order of a sequence we have  holds true for the case when .
Hence utilizing this above fact we have
Also the Chapman Kolmorogrov equation can be written as , remember the summation is being done for only those  which are elements/members of .

Now (i)  and (ii)  iff  and
Hence we have:

, which in matrix notation is , i.e.,