Theorem 1.5

Let  be an arbitrary but fixed state, then
(i) is transient iff the series  is convergent (i.e., ) and in this case  is convergent for each .
(ii)  is recurrent iff the series  is divergent (i.e., ) and in this case  is convergent for each  which communicates with .

Proof of Theorem 1.5
Let  be a recurrent state and let  be its mean recurrence time, also define  if
(i) If  is periodic, then  and , where we already know that
(ii) If  has a period , then  and for each state  which communicates with  , where  is the smallest value of  for which

Absorbing Markov Chains

Let us assume a hypothetical example where we have a Markov chain such that all the persistent states () of this Markov chain are absorbing, while  set of states of this same Markov chain are transient. We rearrange the states in such a way (no one stops us from doing this as there is no set pattern in which the states will be reached in the stochastic process), such that we have the transition probability matrix as give: , and here  is a sub-matrix which corresponds to the transition among states, , such that , while  the unit matrix corresponds to the transition among state, , such that  and  is any matrix. Then calculate .

Example for better illustration: consider we have the transition matrix as given