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[markov chain] prove that probability equals 6/pi²

  1. Jan 20, 2012 #1
    1. The problem statement, all variables and given/known data
    attachment.php?attachmentid=42880&stc=1&d=1327088638.gif

    2. Relevant equations
    N/A

    3. The attempt at a solution
    I'll shortly explain what my reasoning is so far, but please ignore it if it comes across too jumbled:
    ---
    Let P denote the markov matrix associated with this problem, then I think I was able to argue that the probability that is asked for is equal to [itex]1- \sum_{n=1}^{+\infty} P^n(0,0)[/itex] where [itex]P^n(0,0)[/itex] denotes the element in the first row of the first column of the n-th power of the Markov matrix.

    And I then wanted to calculate [itex]P^n(0,0)[/itex] for every n by trying to find a pattern in [itex]P^1(0,0)[/itex], [itex]P^2(0,0)[/itex], [itex]P^3(0,0)[/itex], etc. I think I found one: define [itex]t_n = \left(t_{n-1} + \left( \frac{n+1}{n} \right)^2 \prod_{k=1}^{n+1} x_k \right) x_n[/itex] (with [itex]t_0 = x_1[/itex]) with [itex]x_n = p_{n,n-1}[/itex], then I think [itex]\sum_{n=1}^{+\infty} P^n(0,0) = \sum_{n=0}^{+\infty} t_n[/itex]. But it seems near impossible to prove that 1 minus this expression equals [itex]\frac{6}{\pi^2}[/itex] so I'm probably way off track...
    ---

    Any better suggestions? How would you approach this problem instead?
     

    Attached Files:

  2. jcsd
  3. Jan 20, 2012 #2

    micromass

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    Just an idea. In some circumstances (if [itex]\|P\|\leq 1[/itex]) we have

    [tex]\sum_{n=0}^{+\infty}{P^n}=(I-P)^{-1}[/tex]

    So all you need to do know is to invert I-P...
     
  4. Jan 20, 2012 #3
    *tries to invert infinite-dimensional matrix*
     
  5. Jan 21, 2012 #4

    Ray Vickson

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    I found the notation confusing at first, but now I see that what you want to prove is that the chain is transient, with Pr{return to 0} = 1-6/pi^2.

    The first order of business is to establish that the chain is transient, with P{return to zero} < 1. Since you have a discrete-time birth-death process, quite a lot is known or knowable about your system. For example, look at the 1995 paper by Van Doorn, freely downloadable from
    http://journals.cambridge.org/downl...21a.pdf&code=f73435367827c0b479829c68c457666d
    (Google search on "birth-death process+discrete time", and look at the 4th entry "Geometric Ergodicity ... "). In particular, Theorem 2.1 may be a starting point. As for the problem of actually computing the return probability, all I can suggest is that you look at the first-passage-probability equations (for end-state 0) and try to solve them, perhaps using z-transform techniques, or something similar.

    Good luck.

    RGV
     
    Last edited: Jan 21, 2012
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