[markov chain] prove that probability equals 6/pi²

1. Jan 20, 2012

nonequilibrium

1. The problem statement, all variables and given/known data

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:
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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 $1- \sum_{n=1}^{+\infty} P^n(0,0)$ where $P^n(0,0)$ 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 $P^n(0,0)$ for every n by trying to find a pattern in $P^1(0,0)$, $P^2(0,0)$, $P^3(0,0)$, etc. I think I found one: define $t_n = \left(t_{n-1} + \left( \frac{n+1}{n} \right)^2 \prod_{k=1}^{n+1} x_k \right) x_n$ (with $t_0 = x_1$) with $x_n = p_{n,n-1}$, then I think $\sum_{n=1}^{+\infty} P^n(0,0) = \sum_{n=0}^{+\infty} t_n$. But it seems near impossible to prove that 1 minus this expression equals $\frac{6}{\pi^2}$ so I'm probably way off track...
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Any better suggestions? How would you approach this problem instead?

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2. Jan 20, 2012

micromass

Staff Emeritus
Just an idea. In some circumstances (if $\|P\|\leq 1$) we have

$$\sum_{n=0}^{+\infty}{P^n}=(I-P)^{-1}$$

So all you need to do know is to invert I-P...

3. Jan 20, 2012

nonequilibrium

*tries to invert infinite-dimensional matrix*

4. Jan 21, 2012

Ray Vickson

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