$\lim_{n\to\infty,\;n\not\in E} \ell(n)n^{1-\beta}L^n v=d_\beta\int_X v\,d\mu$ pointwise on $X'$.

    Moreover, if $v\ge0$, then $\liminf_{n\to\infty} \ell(n)n^{1-\beta}L^n v=d_\beta\int_X v\,d\mu$ pointwise on $X'$.

-   If $\beta\in(0,\frac12)$, then $L^nv=O(\ell(n)n^{-\beta})$ uniformly on compact subsets of $X'$.

\(i) It is known that the asymptotic behaviour of $L^n$ might be complicated. Chung [@Chung Section I.10] gives an example of a null recurrent Markov chain for which the ratio of $n$-step transition probabilities $p_{ij}^n/p_{k\ell}^{(n)}$ has no limit as $n\to\infty$ (the regular variation assumption on the return time probabilities is violated). Hajian & Kakutani [@HajianKakutani64] (see also [@Aaronson Proposition 1.4.7]) prove that there always exist [*weakly wandering*]{} sets $W$ of positive measure. For such sets, $\int_WL^n1_W\,d\mu=0$ for infinitely many $n$. (Such indicator functions $1_W$ do not lie in our class of observables $v=\xi u$).

\(ii) In the special case of the family , Theorem \[thm-AFN\](a) recovers the result of Thaler [@Thaler00] for $\beta=1$ and the cases $\beta<1$ are new. Parts (b,c) are probably not optimal for  but are the best one can expect in the general setting, see Remark \[rmk-Gouezel\].

\(iii) In addition to yielding convergence results for $L^n$ (rather than $\sum_{j=1}^n L^j$) our methods also cover much wider classes of observables than was previously possible. An indicative example is 