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In my textbook (Dobrow) p.89. The stationary distribution for the Ehrenfest model with transtition matrix for $\{0,1,...,N\}$

$$P_{ij}=\begin{cases}\frac{i}{N}, &\text{if }j=i+1\\ \frac{N-i}{N}&\text{if }j=i-1\\ 0 &\text{otherwise} \end{cases} $$

is computed and shown to binomial with parameters N and 1/2. It then says without justification that there is no limiting distribution. I would like to know why this is so. Thanks.

What I've done so far:

I think I need to show that $lim_{n \to\infty}P^n_{0,0}$ does not exist. (It is also sufficient to show it for any other choice of states)

I know that $P^{2k+1}_{0,0}=0$ for all k>0. I need to show that $lim_{k \to \infty} P^{2k}_{0,0}$ either oscillates or converges to something else. I am having trouble with this. Working on small cases. $P^4_{0,0}=\left(P^2_{0,0}\right)^2+P^2_{0,0}P^2_{1,1}$. $P^6_{0,0}=\left(P^2_{0,0}\right)^3+P^4_{0,0}P^2_{0,0}+P^4_{0,0}P^2_{2,2}$

jlammy
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johnson
  • 482

1 Answers1

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$P^k_{0,0} = 0$ for any and all $k$ because in the Ehrenfest model, marbles (or fleas) can only exchange urns (or dogs) and can only increase or decrease by one. The transition matrix is linked, and no matter many times we propagate the transition matrix $P_{ii} = 0$ for any $i$.

Ehrenfest Model Transition Matrix

dtg67
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