# 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. – Xi'an Apr 2 '18 at 8:05
• @Xi'an Edited. Hope that helps – Physkid Apr 2 '18 at 8:09
• What is $q(i,j)$? Not a probability, obviously. And please add the reference to the quote you reposted. – Xi'an Apr 2 '18 at 8:11
• If that is not a probability, then, I will struggle to know. The author has not defined what that means. – Physkid Apr 2 '18 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.