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?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 finitestate 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. 

