literature to learn more on ergodic harris recurrent chains with an atom

Question

I'm trying to learn more on the topic mentioned in the title. Namely I'd like to get more information on the behavior of the boundary terms. ie if I decompose sum of my chain (suppose it's real-valued) $X_1 + X_2 + \ldots + X_n = D + T_1 + T_2 + \ldots + T_k + W$ where D is the time until we get to the atom $\alpha$ for the first time, $T_i$ is the i-th trip starting and ending in $\alpha$ and W is the tail of the sum after last visit to $\alpha$ then I'd like to know how 'small' are D and W when divided by n. I know they kinda converge in probability to zero and I think I know how to prove it (not sure if it's very rigorous though) but I'd also like to know, for example, that $\frac{1}{n} E (D^2 + W^2) \rightarrow 0$. I tried reading Meyn and Tweedie, but it was rather cumbersome and quite difficult. Can anyone suggest some reading where this is explained as clearly as possible? Doesn't have to be a very general setting, we can assume the chain has an atom so we don't have to add an artificial one, we can make any assumptions we want that will make it easier, I'm just interested in getting a 'feeling' for it, I'd like to better understand terminology and basic concepts.