the family  where $\beta\in(\frac12,1]$. There is a constant $C>1$ such that $C^{-1}x^{-\frac{1}{\beta}}\le h(x)
\le Cx^{-\frac{1}{\beta}}$. Consider observables of the form $v(x)=x^q$. Whereas the results of [@Thaler95; @Zweimuller00] yield uniform convergence of $\sum_{j=1}^n L^jv$ (and of $L^nv$ when $\beta=1$) on compact subsets of $(0,1]$ if and only if $q\beta\ge1$, our results apply for $(1+q)\beta>1$.

\(iv) An immediate consequence of Theorem (a) is that if $\beta\in(\frac12,1]$, $v$ is of the required form $v=\xi u$, and $w\in L^1(X)$ is supported on a compact subset of $X'$, then $\lim_{n\to\infty}m(n)\int_X v\,w\circ f^n\,d\mu=
d_\beta\int_Xv\,d\mu\int_X w\,d\mu$.

\(v) The situation changes considerably if $\int_X v\,d\mu=0$. We have the following result which has no counterpart in standard renewal theory, though Gouëzel [@Gouezel04] proves an analogous (and equally unexpected) result in the case of finite ergodic theory:

\[thm-zero\] Suppose that $f:X\to X$ is a topologically mixing AFN map with regularly varying tail probabilities, $\beta\in(0,1)$. Suppose that $v$ is of bounded variation and is supported on a compact subset of $X'$. If $\int_X v\,d\mu=0$, then $L^nv=O(\ell(n)n^{-\beta})$ uniformly on compact subsets of $X'$.

#### Second order asymptotics and rates of mixing

In certain situations, including the family , the tail probabilities satisfy $\mu(\varphi>n)=cn^{-\beta}+O(n^{-q})$ for some $q>1$, $c>0$. It is then possible to 