Theorem 1.7

Let a finite Markov chain with state space  be also martingle. Then

(i)  for

(ii)  for

(iii)  for

Proof of Theorem 1.7

Before we go through this simple proof we illustrate the concept of a martingle. Now a stochastic process  is said to be martingle process if (i)  for all  and (ii).Now taking expectation we have , i.e.,  

(i) Let  be independent random variables with mean 0, and let , then  is a martingle
(ii) Let  be independent random variables with mean 1, and let , then  is a martingle

Now as this is a martingle as well as a Markov chain with transition probability matrix P, then we would definitely have ,, then it means that

, i.e.,

Now  is satisfied for  iff  and for  iff . Thus if a finite Markov chain is also martingle, then its terminal states are absorbing.

Assuming that there are no further closed sets we get that the interior states, 1,2,…, (l-1) are all transient, hence  for  and similarly we have   for  and  for