# Markov limit expectation

#### JesperHP

##### TS Contributor
I need to find a limit involving a discrete finite state markov chain with absorbing state.

It is given that:

$$q(x,c) = r(x) + \beta E_{c'\lvert c}[q(x,c')]$$

so my thought are:

$$q(x,c') = r(x) + \beta E_{c''\lvert c'}[q(x,c'')]$$

such that :

$$q(x,c) = r(x) + \beta r(x) + \beta^2 E_{c''\lvert c}[q(x,c'')]$$

and repeating infinitely to get limit $$\frac{r(x)}{1-\beta}$$

but I do not know how to prove this ...

my guess is also that since there is a single absorbing state then:

$$\lim_{n \rightarrow \infty} E_{c^n\lvert c}[q(x,c^n)] = q(x,a)$$

with $$a$$ being the "absorbing" value if that makes sense.