# Approximating a hitting time for some state using the stationary distribution?

Provided a random walk on a bounded interval, with step probabilities, $p$ and $q$ and a stationary distribution $\pi$, how "bad" of an approximation is to assume that the hitting time for a position $x=j$ is proportional to $\approx \pi(j)^{-1}$? Can the error be bounded?

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the stationary distribution contains no dynamical information, so I don't see how it could be used to approximate the hitting time. – Carlo Beenakker Jan 8 '13 at 13:30

Let's abstract from the random walk formulation, as you first have to specify what do you mean by the random walk on a bounded interval. In any case, it will be an example of an irreducible finite-state Markov Chain $\Phi$. Let $\pi$ be the invariant probability distribution of $\Phi$. We know that for any state $j$: $$\pi(j) = \frac{1}{\mathsf E_j[\tau_j]} \ ,$$ see e.g. here, so $$\mathsf E_j[\tau_j] = \frac{1}{\pi(j)}$$ and it does make sense to estimate $\tau_j$ as its expected value. The quality of such approximation may be found e.g. using the standard deviation of $\tau_j$. Please, tell me if you are interested in any further details.

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$C=1$ – Ori Gurel-Gurevich Jan 17 '13 at 18:36
@Ori: thanks ${{}}$ – Ilya Jan 18 '13 at 11:57