2
$\begingroup$

I am working though the book of J. Norris, "Markov Chains" as self-study and have difficulty with ex. 2.7.1, part a.

The exercise can be read through Google books. My understanding is that the probability is given by (0,i) matrix element of exp(t*Q). Setting up forward evolution equation leads to differential difference equation which I was hinted admits no closed form solution (see my question on mathoverflow).

I obviously need an nudge in a right direction, and the exercise must admit a simple solution. Any directions are much appreciated.

2 Answers 2

2

It seems you misread the question. One is not interested in the probability of the event $[X_t=i]$ for a given time $t$ but in the probability of the event that there exists a time $t$ such that $X_t=i$.

Here is a nudge...

One approach is to consider the sequence $(Y_n)_n$ of the successive different states visited by $(X_t)_t$. You could try to show that $(Y_n)_n$ is a discrete Markov chain, to compute its transition probabilities (these are very simple and, hint, they do not depend on the numbers $q_i$) and, finally, to compute the probability that there exists an integer time $n$ such that $Y_n=i$.

  • 0
    @Sasha how did you find the probability P_0(T_i? I dont see how to do this.2014-12-14