Module 1:Concepts of Random walks, Markov Chains, Markov Processes
Lecture 3:Markov Chains
Corollary based on Theorem 1.3
Is the following holds true, i.e., , and if is recurrent, then is alsorecurrent.
Proof of corollary based on Theorem 1.3
If , then there exists , such that we have and . Now if , which is arbitrary, we can obtain , which we utilize to sum up, which leads us to . Now if diverges, so does . Now we already know that a state is recurrent iff , hence it would immediately lead us to the fact that is recurrent if is recurrent.
, and if 
is 
is also
, then there exists 
, such that we have 
and 
. Now if 
, which is arbitrary, we can obtain 
, which we utilize to sum up, which leads us to 
. Now if 
diverges, so does 
. Now we already know that a state 
is 
, hence it would immediately lead us to the fact that 
is 
is 
