MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.