Timeline for A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal
Current License: CC BY-SA 4.0
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| Mar 1 at 19:55 | comment | added | user133281 | $B^2$ 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 | |
| Mar 1 at 19:25 | comment | added | Dieter Kadelka | Do you know an interpretation of $B^2$ in the context of continuous-time MC? | |
| Mar 1 at 18:02 | history | edited | user133281 | CC BY-SA 4.0 |
added 24 characters in body
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| Mar 1 at 17:29 | history | asked | user133281 | CC BY-SA 4.0 |