Reccurent vs. Transient in Stochastic proccess.

[MathematicalPhysicist]

I have a few quetions which I'm trying to answer, hope someone can hint me to the right answer.
1.Prove/disprove that there doesn't exist a Markov Chain with transition matrix which all of its entries are positive and which it has infinite transient states and infinite reccurent states.
2. There's given a Markov chain which its states are the integers, and for each i, the transition probabilities are given by:
$$P_{i,i-1}=P_{i,i}=P_{i,i+1}=1/3$$
Can we sort out the states of the chain (as reccurent or transient) with the help of the strong rule of the large numbers?

For 1, I think that such a matrix obviously should be infinite, the problem is that I think that in this case each states would be reccurent and not transient so the statement is valid.
Not sure how to rigourosly write it though.

For 2, I've computed the expectation value and it's value is 0, Now if we were to use the strong rule, then it means that the culminated averages of the displacement approches zero, which means that: $$\frac{\sum_{i=0}^{n}X_i}{n}\rightarrow 0$$ as $$n\rightarrow \infty$$, which means that: $$\sum_{i=0}^{\infty} X_i< \infty$$ but not necessarily zero, so we can't know for sure if 0 is transient or reccurent, if this sum was zero, then it would be reccurent from the strong law because we visit zero infinite times, and if it were other value then it would be transient.

Am I way off here, or on the right track?

Thanks in advance.

 

[mmwave]

Hello,

Thank you for your questions. I will try to provide some guidance and hints to help you find the answers to your questions.

1. To prove or disprove the existence of a Markov Chain with the given conditions, you can start by considering the definition of recurrent and transient states. A state is recurrent if it is possible to return to that state infinitely many times, and it is transient if there is a non-zero probability of never returning to that state. With this in mind, think about the implications of having an infinite number of recurrent and transient states in a Markov Chain. Can you come up with an argument for why this may not be possible?

2. For the second question, you are on the right track. The strong law of large numbers states that the average of a sequence of random variables will converge to the expected value as the number of terms in the sequence increases. In this case, the expected value is 0, but as you mentioned, this does not necessarily mean that the sum of the sequence will be 0. However, keep in mind that the strong law of large numbers only applies to independent and identically distributed random variables. Can you determine if the sequence of displacements in this Markov Chain satisfies these conditions?

I hope this helps you in finding the answers to your questions. Remember to always think critically and carefully consider the definitions and properties of Markov Chains when approaching these types of problems. Good luck!