2
$\begingroup$

Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as well as a reflecting boundary), and $x_L$ is absorbing.

For each position in the walk, one of $N$ jump probabilities ($N \leq L$) is assigned (forward - $p_k$, backward - $(1-p_k)$) from a set $P$, where $(p_1, p_2, ..., p_N) \in P$. However, we do not have knowledge about these assignments. All we are provided with is a set $M$, $(m_1, m_2, ..., m_L) \in M$, of mean occupancy values for each position in the one-dimensional lattice, $(x_0, x_1, ..., x_L) \in L$.

Now, provided access to $M$, to what extent can we find the values for the set of jump probabilities, $(p_1, p_2, ..., p_N) \in P$ (as defined above), for each position in the lattice, $x_k$? Can we guarantee a unique solution by placing certain restrictions on the finite set of jump probabilities $P$?


(Note - This is the reverse formulation of an earlier question I asked about computing mean occupancy for sites in the one-dimensional random walk from assigned jump probabilities. See below for the earlier question.)


Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as well as a reflecting boundary), and $x_L$ is absorbing.

For each position in the walk, we assign one of $N$ jump probabilities (forward, $p_k$, backward, $(1-p_k)$) from a set $P$, where $(p_1, p_2, ..., p_N) \in P$.

For the duration of the random walk, until the absorbing target $x_L$ is reached, what is the mean occupancy of the a given position in the one-dimensional lattice, $x_k$? I'm hoping to find an efficient method to compute an exact solution.

$\endgroup$
5
  • $\begingroup$ Is the measure on $P$ uniform? $\endgroup$ Jun 3, 2010 at 20:00
  • 1
    $\begingroup$ If so, consider a given assignment of elements of $P$. The concomitant transition matrix can be constructed straightforwardly (including a "coffin state") and the associated fundamental matrix as well ( books.google.com/… ). This will give you the information you need for that particular assignment. Then average over the assignments from $P$. $\endgroup$ Jun 3, 2010 at 20:06
  • $\begingroup$ Steve, thank you, I appreciate the link. I suppose my question though was - given a 'particular' random assignment of jump probabilities, can one do better than averaging over all the assignments from P to find the mean occupancy for a position $x_k$? Aside from the contribution of a particular initialization state, my intuition was that the mean occupancy would not be the same for all sites. $\endgroup$
    – Rob Grey
    Jun 3, 2010 at 21:31
  • $\begingroup$ Dear Steve, I updated my question to reflect my previous comment. Perhaps it will make a difference? My apologies for that. $\endgroup$
    – Rob Grey
    Jun 3, 2010 at 21:34
  • $\begingroup$ If you have occupancy times of a birth n' death process until hitting some particular state they are geometric, and if you put the process in continuous time they are exponential, and I think you can relate even joint occupancies to values of bessel processes via imbedding them in brownian motion & using something like a Ray-Knight theorem. $\endgroup$
    – mike
    May 11, 2012 at 14:07

0

Your Answer

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

Browse other questions tagged or ask your own question.