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Reccurent vs. Transient in Stochastic proccess.

  1. Mar 26, 2009 #1

    MathematicalPhysicist

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    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:
    [tex]P_{i,i-1}=P_{i,i}=P_{i,i+1}=1/3[/tex]
    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: [tex]\frac{\sum_{i=0}^{n}X_i}{n}\rightarrow 0 [/tex] as [tex]n\rightarrow \infty[/tex], which means that: [tex]\sum_{i=0}^{\infty} X_i< \infty[/tex] 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.
     
  2. jcsd
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