Theorem 1.2

1) If , then

2) If any state  has the value of periodicity as , then there exists an integer N (depending on ), such that for all integers ,

3) If , then  for all n (positive integer) sufficiently large
A Markov chain for which each state has periodicity of one (1) is called aperiodic.

Generating functions

Let us first define the concept of general generating function. In case we have a sequence, , then the generating function, , is defined as  for the case when . Hence in a similar manner the generating function for the sequence , , is given by ,
We know that if we have (i) ,  and (ii) , , then we can write the product of  as given below, i.e.,

, where

Let us identify  with the  and the  with the  and if we compare  for  with  we immediately obtain  for  or  for

One should remember that  for  is not valid for n = 0 and using simple arguments we have

 for

where  is the probability that the first passage from state  to state  occurs at the  transition. Again  , and utilizing , we can easily show that  for  can be written as ,

We say a state  is recurrent iff , i.e., a state is recurrent iff the probability is 1 so that starting at state  it will return to state  after a finite number of transitions. A non-recurrent state is said to be transient.