4
$\begingroup$

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?

$\endgroup$
1
  • 2
    $\begingroup$ the stationary distribution contains no dynamical information, so I don't see how it could be used to approximate the hitting time. $\endgroup$ Jan 8, 2013 at 13:30

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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