# Definition of (contiuous-time) Markov chain transition rate

Suppose that the rate at which a Markov chain leaves state i at some time t is $\lambda_{i}$. I.e., What is the rate at which $X_{t}$ leaves state i.

Then, $\lambda_{i} = \sum_{j\neq i}q\left ( i,j \right )$ is the rate at which a Markov chain leaves a state i for j.

From observation, $\lambda_{i} = \sum_{j\neq i}q\left ( i,j \right )$ looks to be the definition of a marginal probability up to some state j.

Could someone kindly explain to me why there is the case? Why would the rate at which the chain moves be determined by how far is it transition previously? It's bizarre.

The notation $q(i,j)$ is also not clarified nor defined by the author.

• Unclear question: what is the role of $t$? And, given the definition of $\lambda_i$, which is the probability of not staying in $i$, i.e., $1-q(i,i)$, it is not a rate. Commented Apr 2, 2018 at 8:05
• @Xi'an Edited. Hope that helps Commented Apr 2, 2018 at 8:09
• What is $q(i,j)$? Not a probability, obviously. And please add the reference to the quote you reposted. Commented Apr 2, 2018 at 8:11
• If that is not a probability, then, I will struggle to know. The author has not defined what that means. Commented Apr 2, 2018 at 8:28

The notation $q(i,j)$ is also not clarified nor defined by the author.
Looking at that definition you will see that $\lambda_i$ doesn't depend on any past event.