3
$\begingroup$

I apologize if this is too elementary a question, but I have not been able to make much progress.

Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. The matrix $A$ is the generator matrix for some continuous-time Markov chain $M$. Define the matrix $$ B = \lambda (\lambda I - A)^{-1} $$ for some $\lambda > 0$. The matrix $B$ is a stochastic matrix that describes the behavior of $M$ over time intervals of random length $t \sim \text{Exp}(\lambda)$.

Conjecture. The diagonal entries of the matrix $B - B^2$ are non-negative.

The conjecture is not true for an arbitrary stochastic matrix $B$, but in numerical simulations it seems to be true for stochastic matrices of the special form above. I was not able to make much progress toward proving the conjecture, so any ideas are appreciated.

Improve this question
$\endgroup$
2
  • $\begingroup$ Do you know an interpretation of $B^2$ in the context of continuous-time MC? $\endgroup$
    – Dieter Kadelka
    Commented Mar 1 at 19:25
  • $\begingroup$ $B^2$ gives transitions over a time interval of length $t = t_1+t_2$, where $t_i \sim \text{Exp}(\lambda)$. So the conjecture can be rephrased as $$\mathbb{P}(\mbox{in state $i$ at time $t$} \mid \mbox{in state $i$ at time $0$, $t \sim Exp(\lambda)$}) \ge \mathbb{P}(\mbox{in state $i$ at time $t$} \mid \mbox{in state $i$ at time $0$, $t=t_1+t_2$, $t_1,t_2 \sim Exp(\lambda)$}).$$ Not sure how helpful this is though, it seems like an intuitive statement but it is not true in general that the probability to be in state $i$ at time $t$ (conditional on being in state $i$ at time $0$) decreases $\endgroup$
    – user133281
    Commented Mar 1 at 19:55

0

Reset to default

You must log in to answer this question.